U.S. patent application number 12/818826 was filed with the patent office on 2011-03-17 for controllable miniature mono-wing aircraft.
This patent application is currently assigned to UNIVERSITY OF MARYLAND. Invention is credited to STEVEN GERARDI, JOSEPH PARK, DARRYLL J. PINES, EVAN R. ULRICH.
Application Number | 20110062278 12/818826 |
Document ID | / |
Family ID | 43729544 |
Filed Date | 2011-03-17 |
United States Patent
Application |
20110062278 |
Kind Code |
A1 |
ULRICH; EVAN R. ; et
al. |
March 17, 2011 |
CONTROLLABLE MINIATURE MONO-WING AIRCRAFT
Abstract
Micro/nano mono-wing aircraft with the wing configured as a
winged seed (Samara) is uniquely suited for autonomous or remotely
controlled operation in confined environments for surrounding
images acquisition. The aircraft is capable of effective
autorotation and steady hovering. The wing is flexibly connected to
a fuselage via a servo-mechanism which is controlled to change the
wing's orientation to control the flight trajectory and
characteristics. A propeller on the fuselage rotates about the axis
oriented to oppose a torque created about the longitudinal axis of
the fuselage and is controlled to contribute in the aircraft
maneuvers. A controller, either ON-board or OFF-board, creates
input command signals to control the operation of the aircraft
based on a linear control model identified as a result of extensive
experimentations with a number of models.
Inventors: |
ULRICH; EVAN R.; (FREDERICK,
MD) ; PINES; DARRYLL J.; (CLARKSVILLE, MD) ;
PARK; JOSEPH; (PASADENA, MD) ; GERARDI; STEVEN;
(SALISBURY, MD) |
Assignee: |
UNIVERSITY OF MARYLAND
COLLEGE PARK
MD
|
Family ID: |
43729544 |
Appl. No.: |
12/818826 |
Filed: |
June 18, 2010 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
61218618 |
Jun 19, 2009 |
|
|
|
Current U.S.
Class: |
244/48 |
Current CPC
Class: |
B64C 39/028 20130101;
B64C 2201/028 20130101; B64C 39/024 20130101; B64C 27/00
20130101 |
Class at
Publication: |
244/48 |
International
Class: |
B64C 3/38 20060101
B64C003/38 |
Claims
1. A miniature mono-wing aircraft, comprising: a wing configured in
the shape of a winged seed (Samara), a fuselage member flexibly
coupled to said wing at one end thereof, and a controller unit
operatively coupled to said wing to control said wing orientation
relative said fuselage member, thereby controlling said mono-wing
aircraft displacement.
2. The miniature mono-wing aircraft of claim 1, further comprising
at least one propeller attached to said fuselage member in
proximity to one end thereof and actuated by said controller
unit.
3. The miniature mono-wing aircraft of claim 2, wherein an axis
c.sub.x extends along said fuselage member, wherein an axis c.sub.y
is a pitch axis of said wing, wherein said axis c.sub.x is
substantially perpendicular to said axis c.sub.y, wherein an axis
c.sub.z of the aircraft rotation extends in perpendicular to a
plane formed by said axes c.sub.x,c.sub.y, and wherein a said at
least one propeller rotates in a plane positioned in a
predetermined angular range relative said axes c.sub.x, c.sub.y,
c.sub.z.
4. The miniature mono-wing aircraft of claim 3, wherein said
predetermined angular range is -90.degree.-+45.degree. in a plane
formed by said axis c.sub.x, c.sub.z, and wherein said
predetermined angular range is -45.degree.-+45.degree. in a plane
formed by said axis c.sub.x, c.sub.y.
5. The miniature mono-wing aircraft of claim 3, further coupling a
second propeller rotating around a rotational axis extending in
opposite direction of a rotational axis of said at least one
propeller.
6. The miniature mono-wing aircraft of claim 2, further comprising:
a servo mechanism coupled to said wing at said one end thereof,
wherein said controller unit is operatively coupled to said servo
mechanism to control the orientation of said wing.
7. The miniature mono-wing aircraft of claim 6, wherein said
controller unit generates a first input control signal
corresponding to a collective pitch command for said wing based on
said aircraft vertical position and heave velocity, and couples
said first input control signal to said servo mechanism.
8. The miniature mono-wing aircraft of claim 6, wherein said
controller unit generates a second input control signal
corresponding to a rotation speed of said at least one propeller
and couples said second input control signal to said at least one
propeller to control a throttle of said aircraft.
9. The miniature mono-wing aircraft of claim 2, wherein said
controller unit is selected from a group consisting of an on-board
control unit secured at said fuselage, and an off-board control
unit operatively coupled to said on-board control unit, wherein
said on-board control unit includes a receiver receiving input
control signals generated to control the operation of said aircraft
in a desired manner.
10. The miniature mono-wing aircraft of claim 1, wherein said wing
has the area centroid thereof located between the geometric center
line and the tip of said wing.
11. The miniature mono-wing aircraft of claim 2, further including
a motor attached to said fuselage member and rotating said at least
one propeller.
12. The miniature mono-wing aircraft of claim 1, wherein said wing
has a leading edge and a trailing edge, and wherein said leading
edge has a larger vertical cross-section than said trailing
edge.
13. The miniature mono-wing aircraft of claim 2, wherein said
fuselage member is attached to said wing in proximity to the center
of mass thereof.
14. The miniature mono-wing aircraft of claim 12, wherein for the
optimal descent, a span of said wing along said trailing edge
thereof is approximately 0.168 m, and the width of said wing
between the leading and trailing edges is approximately 0.048
m.
15. The miniature mono-wing aircraft of claim 7, wherein the heave
velocity w(t) response of said miniature mono-wing aircraft to said
first input control signal is described by .omega. ( t ) = - Z
.theta. 0 Z .omega. .theta. 0 ( 1 - Z .omega. t ) , ##EQU00025##
where .theta..sub.0 is an input collective pitch control signal,
Z.sub..theta..sub.0 is a collective pitch stability derivative, and
Z.sub..omega. is a heave stability derivative.
16. The miniature mono-wing aircraft of claim 7, wherein for said
first input control signal described by:
.theta.=.theta..sub.0+.theta..sub.1c cos .psi.+.theta..sub.1s sin
.psi., said wing resulting orientation is described by:
.beta.=.beta..sub.0+.theta..sub.1c
cos(.omega.-.pi./2)+.theta..sub.1s sin(.psi.-.pi./2), wherein
.theta. is a wing input control pitch signal, .theta..sub.0 is an
input control collective pitch, .theta..sub.1c is an input control
pitch of the wing causing forward motion of the aircraft,
.theta..sub.1s is an input control pitch of the wing causing
lateral motion of the aircraft, .psi. is an azimuth of the wing,
.beta. is the wing response flap to said wing input control signal,
and .beta..sub.0 is the mean flap angle response of said wing.
17. The miniature mono-wing aircraft of claim 1, wherein said
controller unit includes a control unit selected from a group
consisting of: open-loop controller, closed-loop controller, and a
feedback control unit, said feedback control unit including a
Proportional-plus-Derivative Plus Integral (PID) controller.
18. The miniature mono-wing aircraft of claim 7, wherein a
relationship between said first input control signal and a
resulting reaction of said wing is described by a linear control
model.
19. A method of operating a miniature mono-wing aircraft,
comprising the steps of: flexibly attaching a single wing to a
fuselage member, said single wing being configured in a shape of a
winged seed (Samara), securing at least one propeller to said
fuselage member, said propeller rotating in a plane to oppose a
torque created about a longitudinal axis of said fuselage member,
and controlling operation of said miniature mono-wing aircraft by
applying respective input control signals to said wing and said at
least one propeller in a predetermined fashion, including the steps
of: (a) for controlling vertical displacement of said miniature
mono-wing aircraft, actuating said at least one propeller to a
predetermined rotation speed, thereby causing said aircraft to spin
around an end of said wing, thus resulting in a vertical upside
motion of said aircraft, (b) applying said respective input control
signals to said wing to control orientation thereof, thereby
controlling a direction of said aircraft displacement, said
respective control input signals corresponding to a collective
pitch of said wing, (c) controlling horizontal displacement of said
aircraft by applying said respective input control signals to said
wing and said at least one propeller in an impulsive fashion for
timed variations in said wing orientation and rotational speed of
said at least one propeller at desired points of said aircraft
rotational trajectory, and (d) controlling the turning motion of
said aircraft by applying said respective input control signals
corresponding to collective pitch to said wing in sustained
fashion, thereby changing the turn radius of the flight path.
20. The method of claim 19, further comprising the step of:
operating said aircraft in the autorotation mode thereof for a
descent from a predetermined altitude.
Description
RELATED APPLICATION
[0001] This Utility patent application is based on U.S. Provisional
Patent Application 61/218,618 filed on 19 Jun. 2009.
FIELD OF THE INVENTION
[0002] The present invention relates to micro air vehicles, and
more in particular, to a micro/nano unmanned aerial system
employing a single wing configured as a winged seed (Samara).
[0003] In overall concept, the present invention relates to a
miniature mono-wing rotating aircraft which mimics the passive
transit of the winged seed which is capable of a steady hovering
flight, as well as of the autorotation mode of operation that
permits a slow descent of the aircraft in a manner similar to that
of the maple winged seed (Samara), and may be maneuvered for
desired vertical, horizontal and turning displacements.
[0004] The present invention also relates to a micro/nano unmanned
aerial system which includes a single wing flexibly attached to a
fuselage member and a propeller attached to the fuselage member and
controllably rotatable in a predetermined plane to cause a stable
rotational motion of the aircraft. A control unit (ON-board and
OFF-board) controls the orientation of the wing and/or rotation of
the propeller in a predetermined manner to attain desired modes of
the vehicle's operational and flight parameters, as well as to
steer the vehicle for desired flight trajectories.
[0005] The invention is also directed to a micro/nano monocopter
which rotates around a vertical axis of inertia and is adapted to
carry an on-board camera for collecting 360.degree. panoramic
detailed three-dimensional images of its surroundings and which is
easily deployable in an area of interest in a cost effective
manner.
BACKGROUND OF THE INVENTION
[0006] In recent years, a new concept of flight has emerged that
encompasses microscale aircraft that are bio-inspired. These highly
maneuverable platforms are capable of hovering flight and are
ideally suited for operation in a confined environment. The
reconnaissance mission envisioned for micro/nano aircraft requires
a high level of autonomy due to the fast dynamics of the vehicle
and the limits associated with communication in the likely areas of
operation (i.e., caves and buildings).
[0007] Aerial systems that satisfy the dimensional constraints
outlined by the Defense Advanced Research Projects Agency (DARPA)
micro air vehicle (MAV) initiative include fixed-wing, rotary wing,
and flapping-wing vehicles. The simplest and most mature of these
platforms are fixed-wing vehicles that boast speed, simplicity, and
well-known dynamics. However, the limitation imposed by forward
flight restricts functionality in cluttered environments which can
be traversed by rotary- and flapping-wing platforms.
[0008] Microscale helicopter linear dynamic system models have been
developed for substantially larger vehicles, including the Yamaha
RMAX helicopter. Despite the growing interest in microscale
helicopter flight, a dynamic model of a single-winged rotorcraft
has not been developed.
[0009] The concept of a single-wing rotating aircraft is known in
the prior art. The first vehicle of this type was flown in 1952 in
the woods surrounding Lake Placid, N.Y. A more recent vehicle was
developed and flown by a team led by Lockheed Martin Advanced
Technology Laboratories. The prototype, called MAVPro, incorporated
an outrunner motor with an 8-in.-diam propeller; it weighted 0.514
kg, rotated at a stable 4 Hz, and could clime to 50 ft with
radio-controlled actuation of a trailing-edge flap. The MAVPro
incorporated the AG38 airfoil and exhibited a rectangular planform
geometry. However, the various single-winged rotating aircraft
developed over the years have made no attempt to use the most basic
mode of transit of the natural Samara, e.g., the autorotation.
Additionally, airfoil cross sections and planform designs have had
no similarity to those found in natural Samaras.
[0010] Samaras, or winged seeds, are the sole method by which
several species of plants propagate their seed. Geometric
configurations for maximal seed dispersal has evolved into two main
classes of seeds, both of which execute autorotational flight as
they fall from the tree, and one of which additionally rotates
about its longitudinal axis. The optimality of the autorotation
heavily influences the population dynamics. The evolution of the
Samara provides a nearly infinite set of feasible autorotation
configurations with each having distinct dynamics.
[0011] Advancements in technologies associated with the sensing and
control aspect of unmanned vehicles has allowed conventional
micro-scaled vehicles to be equipped with real-time systems. The
vast capabilities provided to these small systems are limited by
the battery life and power consumption of the on-board electronics
and actuators. The majority of the power consumed in an aerial
system sustains a desired flight mode. The primary focus of this
flight mode is to negate the effects of gravity.
[0012] A new paradigm is needed, whose focus is the design of a
vehicle with a passively stable primary mode of operation which
requires little or no additional power to attain/maintain this mode
of transit. The natural flight of a Samara, by trading the
gravitational potential energy for rotational kinetic energy which
perpetuates an aerodynamically stable helical descent, is perfectly
suitable for this purpose. However, no known micro-air vehicle has
used this concept so far.
[0013] In addition, the conventional monocopter designs apply
torque to the vehicle with a thrust device slightly offset from the
c.sub.y axis (shown in FIG. 1). In the case of MAVPro, the
propeller spins in the c.sub.y-c.sub.z plane and influences the
stability about the c.sub.y axis. This configuration results in the
propeller fighting the pitch input from the flap and reduces
controllability of the vehicle. A different configuration is
therefore needed which would provide an improved roll
stability.
SUMMARY OF THE INVENTION
[0014] It is an object of the present invention to provide a
miniature (micro/nano) mono-wing air vehicle capable of a high
level of autonomy due to the low power consumption possible due to
the autorotation and vehicle dynamics of Samara-like
configuration.
[0015] It is another object of the present invention to provide a
robotic Samara-based micro air vehicle uniquely suited to
performing covert and reconnaissance missions which is easily
deployable in an area of interest.
[0016] A further object of the present invention is to provide a
robotic Samara micro air vehicle permitting the ON-board and
OFF-board control of flight parameters and trajectory via the
controllable orientation of the wing.
[0017] It is a further object of the present invention to provide a
micro/nano unmanned aerial system using unconventional Samara
inspired planform geometry and airfoil cross-section performing
stable autorotation and capable of landing at terminal velocity
without sustaining damage.
[0018] It is another object of the present invention to provide a
winged seed configured miniature mono-wing aircraft with an
improved wing geometry adapted for optimal descent.
[0019] It is also an object of the present invention to provide a
micro air vehicle using a miniature propeller controllably spinning
in plane selected to increase the roll stability and steady
hovering of the aircraft.
[0020] In one aspect, the present invention is related to a
miniature mono-wing aircraft apparatus in which wing is configured
in the shape of a winged seed (Samara) of a maple tree capable of
steady hovering and autorotation, and which can be maneuvered in
vertical as well as horizontal directions, as well as desired
turning trajectories by controlling of the wing's orientation in
space.
[0021] The vehicle, in addition to the single wing, includes a
fuselage member (also referred to herein as a holder member)
flexibly attached to the wing in proximity to the center of mass
thereof with a propeller coupled to the fuselage member and
controllably rotating around the rotation axis extended at a
predetermined angle relative to the pitch axis of the wing for
creating rotational motion of the aircraft about the vertical axis
of inertia in the most stable fashion.
[0022] The wing is attached to the fuselage member via a servo
mechanism which is controlled to change the orientation of the wing
selectively as required by a desired mode of operation, flight
parameters, and trajectory of flight.
[0023] The holder member (or fuselage) carries batteries and
electronics thereon. Due to the efficiency of the energy
consumption in the autorotation mode of the air vehicle in
question, small batteries will provide a substantial support for
autonomous flight of the vehicle.
[0024] The electronics embedded into the micro air vehicle may
include an onboard microcontroller for autonomy operation. The
embedded electronics further includes a receiver for receiving
input control signals (thrust, wing pitch, coning) from an offboard
controller. The controller includes a processing unit configured in
accordance with an algorithm corresponding to the control model
underlying the operation of the air vehicle in question.
[0025] The control model for robotic Samara air vehicle is based on
a linear model for the heave dynamics in hovering flight which was
identified from data collected during extensive experimentations
using visual tracking system. The linear model describes the
reaction of the vehicle system to a force imposed by a control
input to the wing and/or the propeller used in the system. The
controller unit of the micro/nano Samara aircraft is envisioned
either as a closed- or open-loop controller, and may include a
feedback control unit based on
Proportional-Plus-Derivative-Plus-Integral (PID) control
principles. A model of the wing's pitch input-to-heave velocity
transfer function, as well as other input/output relationships,
have been identified, including heave velocity and altitude changes
based on collective pitch control input signal and stability of the
system.
[0026] An optimal design of the wing permitting the slowest descent
speed has been identified from a plurality of different models and
includes the parameters of the chord geometry as well as dimensions
of the wing, and position of the center of gravity of the wing. It
is preferred that the leading edge of the wing is formed with an
increased thickness relative to the body of the wing and the
trailing edge to attain optional descent.
[0027] In another aspect, this present invention is a method of
operating a miniature mono-wing aircraft. The method includes the
steps of:
[0028] flexibly attaching a single wing to a fuselage member where
the wing is configured in a shape of a winged seed (Samara);
[0029] securing a propeller to the fuselage member to rotate in a
predetermined plane selected to provide a stable roll and hovering
for the rotational motion of the aircraft; and
[0030] controlling the operation of the aircraft by applying
respective control signals (pitch, thrust) to the wing and/or the
propeller at predetermined time intervals.
[0031] Control of the vertical displacement of the aircraft is
performed by actuating the propeller to a predetermined rotational
speed which causes the aircraft to spin around the vertical axis of
inertia resulting in a vertical upside motion of the aircraft.
[0032] In order to control a direction of the aircraft
displacement, the control signals corresponding to the collective
pitch are applied to the wing to control its orientation. In order
to control the horizontal displacement of the aircraft, respective
control signals are applied to the wing and the propeller in an
impulsive fashion for timed variations in the wing orientation and
rotational speed of the propeller at desired points of the aircraft
rotational trajectory. In order to control the turning motion of
the aircraft respective control signals corresponding to the wing
pitch, are applied to the wing in sustained non-impulsive fashion
to change the turn radius of the flight path. The aircraft is also
operated in the autorotation mode to descend from a predetermined
altitude. For this the propeller may be deactuated, or the
propeller rotation rate is kept at a low predetermined level, and
by controlling the orientation of the wing, a controlled descent
mode is obtained.
[0033] These and other objects of this invention will be apparent
upon a reading of the following detailed description of the
invention in conjunction with the Patent Drawings Figures.
BRIEF DESCRIPTION OF THE DRAWINGS
[0034] FIG. 1A is a perspective view of the micro/nano mono-wing
aircraft of the present invention in an enlarged scale for
clarity;
[0035] FIG. 1B is a schematic representation of controller scheme
of the aircraft of the present invention;
[0036] FIGS. 2A-2D show structural and geometric details of the
wing of the aircraft of the present invention;
[0037] FIG. 2A is a representation of the planform of the wing of
the aircraft of the present invention,
[0038] FIG. 2B is a representation of the cross-section of the wing
taken along line A-A in FIG. 2A,
[0039] FIG. 2C is a cross-section of the wing taken along line B-B
of FIG. 2A, and
[0040] FIG. 2D is a representation of retro-reflective marker
placement on the wing;
[0041] FIG. 2E shows a planform geometry of the mechanical Samara
adapted for optimal aerodynamics in accordance with Reynolds number
distribution;
[0042] FIG. 2F presents a diagram of chord geometries of the
mechanical Samara which are given an increasing camber towards the
tip of the wing;
[0043] FIG. 2G is a diagram of the Reynolds number vs. wing
radius;
[0044] FIGS. 3A-3D are representations of the mechanical Samara
wings having different geometries;
[0045] FIG. 4 is a diagram representing the Reynolds number versus
span length of the wing models shown in FIG. 3A-3D;
[0046] FIGS. 5A-5C are diagrams representing the wings planform
geometry and Fourier series approximation versus span length for
A41, B41, and C41 models, respectively;
[0047] FIG. 6 is a schematic representation of the virtual flight
path of the mechanical Samara wing;
[0048] FIG. 7 is a diagram representing the mechanical Samara wing
flight path data recorded by VICON during experimentations;
[0049] FIGS. 8A-8D are diagrams representing Euler angles computed
from flight test data as time history for the models A41, B41, C41,
and D41, respectively;
[0050] FIG. 9 is a schematic representation of the roll (p), pitch
(q), and yaw (r) defined in body fixed coordinate system;
[0051] FIGS. 10A-10D are diagrams representing time history of
roll, pitch, and yaw rates for the mechanical Samara wing for the
models A41, B41, C41, and D41, respectively;
[0052] FIG. 11A is a schematic representation of the wing section
used to compute area centroid;
[0053] FIG. 11B is the schematic representation of the wing motion
showing the radius of precession of the center of mass
r.sub.CG;
[0054] FIG. 12 is a diagram representing the influence of wing
radius, wing surface area, and wing area centroid distance from
center of mass yc on descent velocity;
[0055] FIGS. 13A-13D are diagrams representing roll and pitch with
95% confidence interval for the models A41, B41, C41, and D41,
respectively;
[0056] FIGS. 14A-14D are diagram representations of the Fourier
series estimation of TSA roll and pitch flight data for the models
A41, B41, C41, and D41, respectively;
[0057] FIG. 15 is a free body diagram of the mono-wing aircraft of
the present invention;
[0058] FIGS. 16A-16D are schematic representations of the mono-wing
aircraft of the present invention showing roll (p), pitch (q) and
yaw (r) definitions for body-fixed coordinate system B; where FIG.
16A is a front view of the aircraft, FIG. 16B is a side view of the
aircraft, FIG. 16C is a top view of the aircraft, and FIG. 16D is a
perspective view of the aircraft of the present invention;
[0059] FIG. 17 shows schematically forces acting on the element of
a flapping robotic Samara wing;
[0060] FIG. 18 is a block diagram of the open-loop control
setup;
[0061] FIGS. 19A-19B are diagrams representing the time history of
the heave velocity (FIG. 19A) and control input pitch (FIG. 19B)
for the Samara I model;
[0062] FIG. 20 is a block diagram of the ground control station
(closed loop) of the system of the present invention;
[0063] FIGS. 21A-21B are diagrams showing, respectively, magnitude,
phase and coherence for onboard and offboard data collection for
Samara I (FIG. 21A) and Samara II (FIG. 21B) models,
respectively;
[0064] FIGS. 22A-22B are diagrams representing magnitude and phase
versus frequency for Samara I and Samara II models,
respectively;
[0065] FIGS. 23A-23B are diagrams representing the results of error
analysis for Samara I and Samara II models, respectively;
[0066] FIGS. 24A-24B are diagrams representing altitude (Z), the
heave velocity (w) and change in response to a perturbation of
heave velocity for Samara I model;
[0067] FIG. 25 is a diagram representing aircraft heave response to
an input of collective pitch;
[0068] FIG. 26 is a block diagram of the feedback control loop;
[0069] FIGS. 27A-27B are diagrams showing height and heave velocity
(w) for Samara I (FIG. 27A) and Samara II (FIG. 27B) responsive to
the PID control;
[0070] FIG. 28 is a schematic representation of modeling the
aircraft of the present invention as a rotor with a hinged virtual
body;
[0071] FIG. 29 is a diagram showing definition of angle of attack
and side slip angle in relation to the velocity components;
[0072] FIG. 30 is a diagram representing azimuth angles of the
wing, virtual body and virtual body with respect to the wing;
[0073] FIG. 31 is a schematic representation of the coning angles
with forces acting on an element of a flapping mechanical Samara
wing;
[0074] FIGS. 32A-32D show a flight path (FIGS. 32A-32B) and
corresponding turn radius and turn rate (FIGS. 32C-32D) of the
aircraft of the present invention;
[0075] FIGS. 33A-33C are diagrams showing the pilot commands
(F.sub.p, .theta..sub.0) in FIG. 33A, resultant velocity components
(u, .omega.) and rotation rate (.OMEGA.) in FIG. 33B, and resultant
flight path and radius of curvature in FIG. 33C;
[0076] FIG. 34 shows diagrams representing equation-error model fit
to perturbation data (high velocity components and rotation
rate);
[0077] FIG. 35 represents diagrams for output-error fit to
perturbation data (velocity components and rotation rate);
[0078] FIGS. 36A-36B represent diagrams of values recorded in
flight for hovering flight (FIG. 36A), and forward flight (FIG.
36B);
[0079] FIG. 37A is a diagram representing modes of the aircraft
operation for throttle (F.sub.p) vs. collective pitch
(.theta..sub.0);
[0080] FIG. 37B is a schematic representation of the pitch angle of
the wing corresponding to FIG. 37A;
[0081] FIG. 38 is a schematic representation of the allowable range
of angular deflection for the fuselage;
[0082] FIGS. 39A-39B are schematic representations of the allowable
ranges of angular deflection for the motor/propeller for angling
the thrust vertically (FIG. 39A) and regarding the center of
gravity (FIG. 39B);
[0083] FIG. 40 is a schematic representation of the alternative
location of the propeller;
[0084] FIG. 41 is a schematic representation of a double
motor/propeller design;
[0085] FIG. 42 is a schematic representation of the acceptable
range for .beta..sub.0; and
[0086] FIG. 43 shows schematically the details of connection of the
servo to the wing and to the fuselage.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
Nomenclature
[0087] A=dynamics matrix [0088] B=control matrix [0089] C=output
matrix [0090] dy=differential element [0091] F.sub.CF=centrifugal
force [0092] F.sub.G=gravity force [0093] F.sub.p=propulsive force
[0094] F.sub.W.sub.D=drag force [0095] F.sub.W.sub.L=lift force
[0096] G.sub.p(s)=plant transfer function [0097] I.sub.x, I.sub.y,
I.sub.z=principal moments of inertia [0098] K=static gain [0099]
K(s)=controller [0100] K.sub.d=derivative gain [0101]
K.sub.t=integral gain [0102] K.sub.p=proportional gain [0103] p, q,
r=rotational velocities [0104] R.sub.BF=transforms fixed frame F to
body frame B [0105] R.sub.xx=input autospectral density [0106]
R.sub.xy=input/output cross-spectral density [0107] R.sub.yy=output
autospectral density [0108] T.sub.pl=time constant [0109] u, v,
w=translational velocities [0110] W(s)=heave transfer function
[0111] {dot over (.omega.)}=heave acceleration [0112] X=state
vector [0113] x, y, z=inertial frame position [0114] Y=control
input [0115] Y.sub.d=reference value [0116] Z.sub..omega.=heave
stability derivative [0117] Z.sub..theta..sub.0=collective pitch
stability derivative [0118] .beta.=coning angle [0119]
.epsilon..sub.i=infinitesimal quantity [0120] .THETA.(s)=collective
pitch transfer function [0121] .theta..sub.0.epsilon.[-1,1]=control
input [0122] .phi., .theta., .psi.=Euler angles [0123] c Chord
length [0124] s Span length [0125] v Signal noise [0126] k
Measurement model [0127] X Deterministic signal [0128] a, b Fourier
series coefficient [0129] r.sub.CG Radius of precession of the
center of mass [0130] Y.sub.C Wing area centroid distance from
center of mass Subscript [0131] i Index of series Superscript
[0132] Estimate
[0133] Referring to FIGS. 1A-1B, 15, and 16A-16D, the robotic
micro/nano mono-wing aircraft 10 includes a wing (airfoil) 12
flexibly coupled to a holder member (also referred to herein as a
fuselage) 14. The flexible connection of the wing 12 to the
fuselage 14 is through the flap hinge 16 which is operable by a
servo controlled mechanism 18. The servo mechanism 18 controls the
orientation of the wing 12 to provide lift and centrifugal force
balance, as well as maneuverability of the aircraft 10 to attain a
desired mode of operation, flight parameters, and flight
trajectory, as will be presented in further paragraphs.
[0134] The wing (also referred to herein as mechanical Samara) 12
is configured similar to a winged seed (Samara) of the maple tree.
The dimensions and geometry of the mechanical Samara wing are
optimized herein for minimal descent rate and optimal aerodynamics
qualities as will be presented infra herein.
[0135] The wing 12 is configured with substantially straight front
edge 42 and a curved rear edge 46 contouring the region 44 of the
wing. Both the rear edge 46 and the region 44 are narrower than the
thick front edge 42 in their respective cross-sections, as will be
presented further herein. As presented in FIGS. 2E-2G, the
mechanical Samara wing design is tuned for its low Reynolds number
environment. The aerodynamics of the robotic Samara is subject to
the scaling of Reynolds number Re, which is the ratio of inertial
to viscous forces, and is a measure of the flow conditions over a
body immersed in a fluid.
[0136] To achieve peak performance it is desirable to have the
lifting surface of a wing operate at its maximum lift-to-drag ratio
(L/D) as this is a measure of the wing's aerodynamic efficiency.
The factors which determine L/D include wing geometry and surface
roughness which influences the flow conditions over a given
airfoil.
[0137] It was found the maximum L/D performance of various airfoils
vs. Reynolds numbers dramatically changed above Re=70,000 for
smooth airfoils, whereas rough airfoils exhibited a steady increase
with Reynolds number and out-performed smooth airfoils below
Re=10.sup.5. The variation of Reynolds number with span length for
a robotic Samara shown in FIG. 2G, crosses this performance
boundary at the out-board section of the wing, e.g., between the
geometric center line and the tip of the wing, as shown in FIG. 2E.
It is therefore advantageous to distribute the wing area (area
centroid Ym) to the wing area between the geometric center line and
the tip 36 of the wing, such that the largest chord sections are
collocated with the largest Reynolds number thereby increasing the
maximum L/D for that wing section. This approach is as well based
on findings from the autorotation experiments which indicated that
an increase in the distance of the area centroid Ym from the center
of rotation leads to an increase in the efficiency of the wing
which is measured by the descent rate. The diagram of chord
geometries shown in FIG. 2F, presents an increased camber towards
the tip of the wing. This is beneficial to increase the angle of
attack with the most lift and the least drag in order to operate
the aircraft in a most efficient aerodynamic state.
[0138] This approach differs from full scale conventional rotor
design which seeks to minimize power losses by creating a uniform
inflow over the rotor disk. This novel approach results in a rotor
blade with large chord sections at a high angle of attack, close to
the center of rotation, and small chord sections at lower angles of
attack farther from the center of rotation. The ideal rotor blade
at full scale results in each chord section operating at maximum
L/D. This novel geometry of the wing also effectively decreases the
skin friction drag.
[0139] The fuselage member 14 has a C-shaped body along its
longitudinal axis C.sub.x shown in FIG. 16D. The fuselage 14
provides a structural support to the wing 12 to change the
orientation, while the wing 12 provides the lift to the aircraft.
The fuselage also carries the embedded electronics 20, source of
propulsion force (for example, motor/propeller), and the power
source.
[0140] A miniature motor 26, for example, a DC brushless 5 W motor,
is secured at a rear end of the fuselage 14. At least one propeller
28 is attached to the rotation axis 30 of the motor 26 to rotate in
the plane C.sub.x, C.sub.z (or in a predetermined angle relative
the same) to create a controlled thrust force F.sub.p. Variations
in the wing pitch which are controlled through the servo control
mechanism 18, as well as changes in the propeller force F.sub.p,
may each contribute into the vertical motion of the aircraft in an
upside or downside direction.
[0141] It is to be understood that the motor/propeller arrangement
illustrated herein is presented as an example only, and any means
capable of producing a propulsion force at the location of the
propeller/motor is contemplated in the present system. The
propeller 28 may be preferably a 3-bladed propeller. However, any
number of propeller blades are envisioned in the subject
aircraft.
[0142] Referring to FIG. 40, the orientation of the propeller 28
may be opposite to that shown in FIGS. 1A-1B. Also, as shown in
FIG. 41, a double motor/propeller setup may be used, as will be
presented infra herein.
[0143] In the configuration presented in FIG. 1A, the aircraft
structure provides a high degree of flexure in the direction
C.sub.z and a high degree of stiffness in the plane of rotation
(C.sub.x, C.sub.y plane). The angle at which the motor 26 is
attached provides a protection from ground impingement on takeoff.
The centrifugal load deforms the structure thereby increasing the
distance of the motor from the center of rotation thus increasing
the applied torque. Since the propeller 28 in the aircraft is
spinning in the C.sub.x, C.sub.z plane, or at a predetermined angle
thereto, and thus opposes the applied torque along the C.sub.x
axis, the aircraft has additional stability in hovering.
[0144] A battery 32 is mounted in the fuselage 14, for example to
the front end 34 thereof (opposite to the end to which the
motor/propeller are attached) to power the operation of the
embedded control electronics 20, servo control mechanism 18, as
well as motor(s) 26 rotating the propeller(s) 28. For example, a
LiPo battery may be used.
[0145] For increased mechanical strength, a landing gear 80 is
provided which includes a carbon fiber rod attached to the
fuselage. The landing gear 80 protects the propeller from impact
with the ground on take-off, and landing. The landing gear 80 may
include wheels which may be attached either to both arms 74 and 76
of the fuselage, or alternatively, a single wheel may be needed in
proximity to the propeller/motor. The location of the landing gear
80 varies with the size and location of the propellers. The landing
gear design is contingent on the surface it is to take-off or land
upon.
[0146] The autopilot system (on-board microcontroller) 38 which is
a part of the embedded control electronics 20 permits the aircraft
to maintain its own vertical position. The horizontal motion of the
aircraft may be controlled through precisely timed variations in
wing orientation and rotational speed of the propeller at desired
points along the aircraft rotation.
[0147] The aircraft 10 is equipped with control electronics 20
which are embedded into the fuselage 14 to control the operation of
the air vehicle 10. Both the ON-board microcontroller and the
OFF-board remote control are contemplated in the subject aircraft
system, based either on the closed-loop or open-loop control
principles to be discussed.
[0148] The ON-board microcontroller 38 as a part of the embedded
control electronics 20 is envisioned in applications which require
full autonomy of the Samara aircraft operation. Alternatively the
remote piloting of the aircraft 10 permits transmission of the
control input parameters from an OFF-board controller unit 22 to
the embedded control electronics 20. The control electronics 20
includes a receiver (or transceiver) 21 for one- or two-directional
communication with the OFF-board control unit 22.
[0149] Referring again to FIG. 1B, the controlling scheme, either
in the ON- or OFF-board arrangement, uses input control signals
(throttle, wing pitch) generated by a processor unit (not shown)
either on-board or off-board. The processor unit operates based on
software capable of generating input control signals depending on a
desired result (mode of operation, flight control model and
trajectory). The relationships between the input and output
parameters have been established for the subject aircraft in the
course of extended experimentations presented further herein.
[0150] The input control signals are coupled to the ON-board
receiver 21. The receiver 21 transmits the processed input control
signals related to the throttle to an electronic speed controller
24 for further control of the motor 26. Also, the receiver 21
couples the data corresponding to the input wing collective pitch
.theta..sub.0 to the servo mechanism 18 to control the orientation
of the wing 12 of the subject aircraft 10 around the pitch axis 31
of the wing 12.
[0151] Upon actuation of the propeller 28, the aircraft 10 spins
itself in a circle around the axis vertical axis shown in FIG. 15
(several rotations per second) and creates lift that causes
inverted aircraft vertical motion along the axis C.sub.z. By
controlling the orientation of the wing 12 through the changes in
the pitch and coning angle of the wing, the trajectory of the
aircraft's motion is controlled.
[0152] If equipped with an onboard camera, the rotational motion of
the aircraft 10 permits collection of a full 360.degree. panoramic
view of surroundings. Detailed 3-D images of the aircraft's
environment may thus be obtained if needed. Being of micro/nano
scale, a large number of aircrafts 10 may be deployed in an area of
interest at a relatively low cost for the intended missions.
[0153] Being based on a natural Samara wing seed configuration, the
aircraft 10 is suited for an autorotational mode of operation for
descent from a specific altitude in power "off" mode of operation,
and therefore thus constitutes an efficient air vehicle system,
permitting autonomous operation.
[0154] The overall weight of the aircraft including the embedded
control electronics (payload), motor, propeller, battery and the
wing in the range of 30-75 grams with the wing span dimensions
being no greater than 16 cm. The vehicle takes off from the ground
and hovers for roughly 10 minutes. The structure supports a useful
payload, for example the embedded electronics package, of
approximately 5 grams.
[0155] Extensive study and experimentation has been conducted for a
plurality of wing geometries to find the optimal geometry of the
wing for minimal descent rate, to produce a quantitative Fourier
series representation of the roll and pitch of the wing, and
establish a relationship between parameters for the prediction of
descent velocity as well as ratios of precession.
[0156] Experimental methology and data processing intended to
characterize the other rotation efficiency and the vehicle
dynamics, as well as the impact of the wing chord length variation
on the descent velocity and altitude dynamics will be described
further herein. This characterization has provided a baseline for
mechanical Samara planform design and provided insights into lift
production of Samara based micro/nano air vehicle. The data aids in
the development and verification of the dynamic/aerodynamic
six-degree-of-freedom mechanical Samara model.
[0157] Physical Model of the Samara Wing
[0158] The models of mechanical Samara were designed by the
planform geometry A41, B41, C41 and D41, shown in FIGS. 3A-3D. The
geometry of the models tested are not simple scaled-up versions of
natural Samaras. Design of the wing (also referred to herein as
mechanical Samara) involves precise placement of the center of
mass, since a poor choice results in a less stable and efficient
autorotation. The low Reynolds number flight regime of the
mechanical Samara (shown in FIG. 4) requires a non-standard airfoil
cross-section which is based on geometric properties observed in
natural Samaras.
[0159] A thick leading edge in a Samara wing results in a 33%
decrease in descent velocity. For this reason, each mechanical
Samara model 40 shown in FIGS. 2A-2D, was designed with identical
cross-sectional properties which include: thick leading edge 42
(about 1.54 mm) followed by about 0.1 mm thick region 44 that
extends to the trailing edge 46.
[0160] Being designed with similar cross-sectional properties, each
model, however, differs from each other in the chord line 50
length. The distribution of mass of each mechanical Samara model
uniquely configured with the thick leading edge 42 followed by the
thin region 44 extending to the trailing edge 46 along the chord
line 50 is required for stable autorotation.
[0161] Each cross section may contain longitudinal stiffeners 48
positioned for example at 0.55 mm, 0.6 mm, and 0.85 mm distance
from left to right on the airfoil cross section, as shown in FIGS.
2A and 2C. Stiffeners span the length of the model 40 and are used
for structural rigidity and integrity of the model.
[0162] The models were designed using CAD software capable of
calculating precisely the model surface area and the location of
the center of mass, as well as overall model mass. These parameters
are held constant over the four different models shown in FIGS.
3A-3D.
[0163] The CAD model can then be exported as a stereo-lithography
file (STL), which is a representation of the Samaras geometry as
approximated by triangles of varying dimension. This data is then
used by the Eden350 software to create the physical prototype. The
tolerances of the machine are 42 .mu.m in the X-Y plane, and 16
.mu.m in the Z-plane. Subjects are built in the same orientation on
the machine to ensure similarity between models. The models were
built from the resin Vero-Black chosen for its high color contrast
with reflective markers used in the experimentations, and its
material properties listed in Table 1.
TABLE-US-00001 TABLE 1 Mechanical Samara material properties,
VeroBlack Property ASTM Unit Value Tensile Strength D-638-03 Mpa
5.070 .times. 10.sup.1 Modulus of Elasticity D-638-04 Mpa 2.192
.times. 10.sup.3 Flexural Strength D-790-03 Mpa 7.960 .times.
10.sup.1 Flexural Modulus D-790-04 Mpa 2.276 .times. 10.sup.3
Density Kg/m.sup.3 1118
[0164] The mechanical Samara tested have the physical properties
listed in Table 2. it should be noted that all mechanical Samara
are planar symmetric, for zero twist. The inertias in Table 2 lack
the final addition of the marker mass, however the final mass does
include this addition.
TABLE-US-00002 TABLE 2 Physical Properties of the mechanical Samara
NAVE A41 B41 C441 D41 1.sup.st Principal I.sub.11 8.99e.sup.-6
6.92e.sup.-6 9.32e.sup.-6 6.93e.sup.-6 Inertia Kg m.sup.2 2.sup.nd
Principal I.sub.22 4.39e.sup.-7 5.05e.sup.-7 4.5e.sup.-7
5.99e.sup.-7 Inertia Kg m.sup.2 3.sup.rd Principal I.sub.33
9.42e.sup.-6 7.41e.sup.-6 9.76e.sup.-6 7.52e.sup.-6 Inertia Kg
m.sup.2 Radius m .1686 .15 .1683. .1359 Mass Kg .0526 .0526 .0526
.0526 Surface Area m.sup.2 1.24e.sup.-2 1.24e.sup.-2 1.24e.sup.-2
1.24e.sup.-2
[0165] The geometry of the planform area for A41, B41 and C41
models is represented through Fourier series approximation of the
chord as it varies with the radius, as presented in FIGS. 5A-5C.
Each of the approximated geometries are reported with the mean
square fit error.
[0166] The coefficients of the Fourier series corresponding to the
above geometries is detailed in Table 3, for the series:
c ^ ( s ) = .alpha. 0 + n = 1 6 .alpha. n cos n .pi. s ( 1 )
##EQU00001##
TABLE-US-00003 TABLE 3 Fourier series coefficient for Samara
geometry Samara Model Fourier Coefficients A41 B41 C41 a.sub.o
.0358 .0414 .0362 a.sub.1 .001 -.0006 -.0071 a.sub.2 -.0084 -.0113
-.0085 a.sub.3 .0032 .0029 .004 a.sub.4 -.0017 -.0024 -.0017
a.sub.5 .0012 .0019 .0019 a.sub.6 -.0011 -.0012 -.0013 MSE 1.20
.times. 10.sup.-4 0.8 .times. 10.sup.-4 3.70 .times. 10.sup.-4
[0167] The Geometry of the NAVE D41 model is calculated from the
schematic provided in FIG. 3D.
Experimental Setup
[0168] To capture the flight dynamics of the mechanical Samara, a
vision based motion capture is considered the most accurate
technique. The VICON vision system was used which eliminates the
need for costly micro-scaled sensor packages. The system collects
data by capturing 2D images of the subject which is fitted with
retro-reflective markers. The VICON system strobes light at the
frame rate of the camera. The light incident on the surface of the
marker returns to its source, reducing errors commonly caused by
interference. The light returned to the lens allows for a quick
computation of the centroid of the marker. Three-dimensional
position is obtained from a least-squares fit of the 2-D camera
observations. The setup of the workspace was tracked in real time
by the VICON system, and allowed the mechanical Samara to fall a
distance of 8 mm before flight data was recorded.
[0169] A simple mechanical grip release mechanism was used to hold
the mechanical Samara at a predetermined angle. This release
mechanism was mounted .about.12 m above the ground, and the
mechanical Samara was released after transient motion has been
eliminated with the physical contact the mechanism makes with the
platform mounted to the ceiling. Each Samara was placed into the
gripper, and hoisted to the ceiling by an attached thread of
monofilament.
[0170] In order to minimize wind disturbances which may effect the
flight dynamics, these experiments were conducted in a room which
had no ventilation. The test facility encompassed two platforms
which provided the area for camera placement. The viewing angle of
the cameras is critical in capture as well as for calibration of
the system. A minimum of three cameras are needed to calibrate the
ground floor plane. This step in the calibration dictates the skew,
if any, of the vertical axis.
[0171] To avoid potential pitfalls from a poorly calibrated
ground-floor-plane, markers were distributed in the area of
interest, and the VICON system took an average of the location of
the markers which enhances the previously calibrated floor-plane.
This calibration results in a least squares estimate of the error
associated with the tracking of each marker. Tracking errors for
the trials were measured by recording data until the subject was
motionless. The resultant measurement characteristics are displayed
in Table 4.
TABLE-US-00004 TABLE 4 Measurement Characteristic Measurement
Symbol Variance Unit Time t -- s Position x, y, z, 0.6128 .times.
10.sup.-3 m Orientation .phi., .theta., .psi. 7.8000 .times.
10.sup.-3 rad Translational Velocity u, v, w 0.2510 .times.
10.sup.-3 m/s Rotational Velocity p, q, r 1.2000 .times. 10.sup.-3
rad/s
[0172] The location of markers placed on the subjects is recorded
by the VICON system prior to the flight testing. This calibration
aids in the proper labeling of reconstructed marker location by
excluding erroneous data that falls outside of the possible range
of marker location for a given subject. This step also ensures the
vision system is able to track the motion of the subject. The
subjects are designed with grooves which trace the outline of the
intended marker location. Any error in marker location is reduced
in the calibration of the subject to the values presented in Table
4. Unlike typical motion capture work which employs spherical
marker geometry, in the subject experimentation, a flat circular
marker that is 0.007 m diameter and 0.0001 m thick is used.
[0173] A marker is placed in the same location on both sides of a
model and represents a single marker location to the visual
tracking system. The conformal markers are intentionally placed on
the subject in locations that are raised from the surface. This
provides the marker with some three-dimensionality which aids in
the vision systems ability to track a subject. The marker discs
were made using a hole punch on 3M 7610 high gain reflective
sheeting. Three markers are placed on the mechanical Samara model
for permitting attitude determination. Marker placement for the
rigid body model is shown in FIG. 2D in VICON inertial coordinates.
A representative schematic showing the virtual flight path of the
mechanical Samara as observed by the vision system can be seen in
FIG. 6. The unprocessed flight data for each of the models, as
recorded by VICON is shown in FIG. 7.
[0174] When working with the VICON software it is necessary to
define a rigid body model which defines the degrees of freedom of
each of the segments of the subject. This helps to ensure only
physically possible solutions are converged upon in the post
processing of a trial. The definition of a rigid body is
accomplished with a fixed body coordinate system, hence the Euler
angles reported for a clockwise descent. It is thus necessary to do
an additional rotation of .pi. about the fixed body X-axis to avoid
the singularity this introduces in the calculation of the
orientation.
Data Reduction and Analysis
Attitude Determination
[0175] The 3D marker position data provides a means of resolving
the orientation of the mechanical Samara in space. The three
markers are sufficient to describe an orthonormal basis from which
the rotation matrix representing the Samaras orientation can be
formed.
[0176] The first vector forms the .sub.j-axis in body frame
coordinate and is the line from the Samara center of mass to the
marker located .about.70 mm in the positive VICON Y-direction. The
remaining basis require an intermediate vector from which to
compute a cross product defining the .sub.k-axis as follows:
.delta. 1 , 2 = M 1 - M 2 ( 2 ) .delta. 1 , 3 = M 1 - M 3 ( 3 ) v =
.delta. 1 , 3 .delta. 1 , 3 ( 4 ) ##EQU00002##
[0177] A schematic detailing the construction of the vector is
shown in FIG. 2D.
[0178] The intermediate vector .delta..sub.1,2 can be normalized
forming the .sub.i body frame axis. The vector v is formed by
normalizing .delta..sub.1,3. This vector is the used to compute the
body frame .sub.k axis. The final body axis is formed in the cross
product of .sub.k and .sub.i. This set forms the orthonormal basis
which defines the orientation of the body with respect to the
inertial frame.
e ~ i = .delta. 1 , 2 .delta. 1 , 2 = a 11 + a 21 + a 31 ( 5 ) e ~
j = e ~ k .times. e ~ i = a 12 + a 22 + a 32 ( 6 ) e ~ k = v
.times. e ~ i = a 13 + a 23 + a 33 ( 7 ) ##EQU00003##
Once the three-dimension marker position is obtained and the basis
set of the mechanical Samara body axes are computed, the Euler
angles can be deduced from the basis .left brkt-bot. .sub.i,
.sub.j, .sub.k.right brkt-bot.. The Euler angles provide a
non-unique set of rotations which can describe the Samaras
orientation, however a singularity arises near .theta.=.+-..pi./2.
Away from the singularity the angles are as follows:
.theta.=arcsin(-a.sub.13) (8)
.psi.=arctan(a.sub.11/a.sub.12) (9)
.phi.=arctan(a.sub.23/a.sub.33) (10)
[0179] To maintain continuity of signs between successive time
steps near the singularity, an effective solution is to set the
angle .psi.=0, and compute the final angle o as:
.phi.=arctan(a.sub.21/a.sub.31) (11)
The computed Euler angles of the tested models A41, B41, C41 and
D41 are shown in FIGS. 8A-8D, respectively.
Attitude Representation
[0180] Subsequent testing of the various mechanical Samara models
provided an insight into the governing dynamics as they varied with
the geometry.
[0181] Attitude representation requires separate basis for the
fixed inertial axes (F) and the body axes which is fixed to the
Samara (B).
F=[{circumflex over (f)}.sub.x,{circumflex over
(f)}.sub.y,{circumflex over (f)}.sub.z] (12)
B=.left brkt-bot. .sub.i, .sub.j, .sub.k.right brkt-bot. (13)
[{right arrow over
(r)}].sub.B=R.sub.1(.phi.)R.sub.2(.theta.)R.sub.3(.psi.)[{right
arrow over (r)}]F (14)
[0182] A schematic detailing the axis of these rotations is shown
in FIG. 9.
[0183] The transformation from the inertial frame to the body frame
is described by three Euler angles. The order of rotation is as
follows: a `yaw` rotation .psi. about the {circumflex over
(f)}.sub.z axis, followed by a `pitch` rotation .theta. about the
new {circumflex over (f)}.sub.y axis, and lastly a `conning`
rotation .phi. about the new {circumflex over (f)}.sub.x axis.
[0184] This rotation sequence is standard for an aircraft.
Rotations in the fixed body frame (B) are orthogonally projected
onto the inertial frame (F), which was derived from finite and
non-commutative rotations. Thus the body angular rates must be
defined separately. The sum of the inner products of each of the
inertial angular rates with the body axis of interest yields the
body angular rates:
p=-{dot over (.psi.)} sin .theta.+{dot over (.phi.)} (15)
q={dot over (.psi.)} cos .theta. sin .phi.+{dot over (.theta.)} cos
.phi. (16)
r={dot over (.psi.)} cos .theta. cos .phi.-{dot over (.theta.)} sin
.phi. (17)
[0185] The roll (p), pitch (q), and yaw (r) time histories for each
of the mechanical Samara model A41, B41, C41, D41 tested are
presented in FIGS. 10A-10D, respectively.
Flight Performance
[0186] The Euler angles o and .theta. display periodic behavior
with varying degrees of shift in each case. This is to be expected
as `pitching` leads `conning` by 90.degree. in full scale
helicopters. This implies the maximum force on the Samara
.theta..sub.max, displaces the Samara maximally .phi..sub.max by
one quarter revolution. This cycle behavior can be seen in FIGS.
8A-8C.
[0187] The scope of flight in this study is characterized by a
nearly constant radius of precession, or more precisely a standard
deviation, .sigma..sub.r.sub.CG, of no more than 6.25% of the mean.
The mean radius of precession is averaged over N periods as
presented in Table 5. An illustration of this concept may be seen
in FIGS. 11A-11B.
TABLE-US-00005 TABLE 5 Time synchronous average parameters for each
of the subjects tested NAVE fHz T sec points N A41 p 7.25 .137 500
8 q 7.25 .138 500 8 B41 p 7.27 .138 384 6 q 7.41 .135 384 6 C41 p
7.32 .137 411 7 q 7.32 .137 411 7 D41 p 5.71 .175 455 6 q 5.71 .175
55 6
[0188] The radius of precession is computed in cylindrical
coordinates where r.sub.CG= {square root over
(x.sub.CG.sup.2+y.sub.CG.sup.2,)} where (x.sub.CG,y.sub.CG) denote
the location of the center of mass of the mechanical Samara model,
or M1 shown in FIG. 2D. This parameter is the conciseness of the
helical descent trajectory, such that a larger r.sub.CG implies a
larger amount of transverse motion.
[0189] The tabulated data shown in Table 6, identify some key
performance parameters, namely, optimal geometry for minimal
descent velocity.
TABLE-US-00006 TABLE 6 Flight performance metrics Metric r.sub.CG,
m .sigma..sub.r.sub.CG, m .sub.CG, m/s .sigma..sub. .sub.CG, m/s
Y.sub.C, m A41 0.07 0.01 -1.60 0.01 64.60 .times. 10.sup.-3 B41
0.16 0.01 -1.70 0.02 59.00 .times. 10.sup.-3 C41 0.27 0.01 -1.50
0.06 69.80 .times. 10.sup.-1 D41 0.40 0.02 -1.80 0.05 56.20 .times.
10.sup.-3
[0190] The mechanical Samara models rotate about the center of mass
or CG, resulting in the majority of the wing area being located on
one side of the CG as shown in FIG. 11A. The distance of the area
centroid of this portion of the mechanical Samaras are calculated
with respect to the CG and shown in Table 6.
[0191] The terminal velocity of natural Samara is a function of two
parameters: the weight of the Samara, and the mass of air it
accelerates. The control volume defining the mass of air
accelerated is defined by Norberg as a flat disk equal in radius to
the wing tip. This definition differs from a definition by Green
which is limited to the surface area of the Samara.
[0192] The mechanical Samara models compared in this study have
identical surface area and weight, for which Green's model predicts
no variation in descent velocity. Norberg's model accounts for the
wing tip radius varying between models which results in a variation
of disk loading. Models A41 and C41 have identical wing tip radii,
therefore Norberg's model predicts no variation in descent
velocity.
[0193] Mechanical Samara models of identical wing loading and disk
loading can be quantitatively compared through observation of the
location of the wing area centroid in relation to the center of
mass, or CG of the Samara. In Table 5 and FIG. 12, the centroid
location farthest from the CG corresponds to the Samara with the
lowest descent velocity. To reconcile the erroneous trends
predicted by the previously mentioned models, a modified disk
loading can be used to predict trends in the terminal velocity of
Samara. This modified disk loading uses the length (Y.sub.C)
between the area centroid and the rotational axis, as the radius of
the flat disk which defines the control volume of air accelerated
by the descending Samara. These relationships are shown in FIG.
12.
Flight Dynamics Analysis
[0194] In applications involving periodic signals it is desirable
to enhance signal to noise ratios in order to extract
representative waveforms. Signal averaging techniques previously
implemented in applications such as structural health monitoring
and optical filter design, are employed in the subject study as a
means of reducing a periodic signal into one discernable waveform.
For a deterministic signal X(t) of period T, a measurement model
can be written as:
k(t)=X(t)+v(t) (18)
where v(t) is additive sensor noise. For (t) measured over N
periods, the ensemble average and ensemble variance can be
approximated as:
n k ( t ) = 1 N n = 0 N - 1 z ( t + nT ) ( 19 ) .sigma. k 2 ( t ) =
1 N n = 0 N - 1 [ k ( t + nT ) - n k ( t ) ] 2 ( 20 )
##EQU00004##
Confidence intervals for estimated parameters can be formulated
from estimation theory. The 95% confidence interval of these
measurements with regard to the signal averages is approximated
as:
k.sub.0.95(t)=n.sub.k(t).+-.1.96.sigma..sub.k(t) (21)
This average is then described for each Samara by Fourier Sine or
Cosine series of varying order. The specific parameters used for
the ensemble averaging are listed in Table 5.
[0195] The roll, pitch and yaw time history for each of the
subjects are displayed in FIGS. 10A-10D. The concatenated signals
are shown in FIGS. 13A-13D. The confidence interval bounds tend to
increase from the mechanical Samara A41 model to the mechanical
Samara D41 model, as shown in FIGS. 13A-13C. Possible causes for
this may include but are not limited to: structural vibrations,
susceptibility to small wind disturbance, and decreased aerodynamic
damping resulting from a change in the center of pressure.
[0196] The number of ensembles averaged is highly dependent on the
settling time of the individual mechanical Samara model, and the
limited drop height. This number could be increased at the cost of
uncertain initial conditions by giving the Samara a pre-spin upon
launch.
[0197] The concatenated roll and pitch flight data may be
represented with a Fourier series allowing a functional
representation of the flight dynamics. The curve fits of the four
models including the 90% confidence interval is shown in FIGS.
14A-14D. The most notable observation is that in steady vertical
descent, roll and pitch are neither constant nor small. This has
substantial implications as a dynamic model was derived based on a
trim state that enforced zero roll and pitch, upon which a number
of vehicle design simulation are based. The waveform indicates a
cyclic variation in the roll rate of roughly .+-.9.5 rads/s for
both A41 and B41 models, whereas C41 model varies by .+-.7.5 rad/s
and D41 model varies from -8 rad/s to 7.5 rad/s. The pitch rate
observed in A41 model is -6 rad/s to 7 rad/s and is similar in
amplitude to B41 model which varies by .+-.7 rad/s. The pitch rates
of both C41 and D41 models appear to have an offset and are not
symmetric about the zero point as observed for A41 and B41 models.
The pitch rate for C41 model may be observed to vary from -5.5
rad/s to 7.5 rad/s and for D41 model the pitch rate varies from 7.5
rad/s to 10.5 rad/s and exhibits a similar offset as that observed
in C41 model. The roll and pitch rates are represented by Eqs.
22-23.
p ^ i ( t ) = .alpha. 0 + n = 1 3 [ .alpha. n cos ( .omega. t ) + b
n sin ( .omega. t ) ] ( 22 ) q ^ i ( t ) = .alpha. 0 + n = 1 3 [
.alpha. n cos ( .omega. t ) + b n sin ( .omega. t ) ] ( 23 )
##EQU00005##
where i=[A, B, C, D]. The yaw rate and curve fit are neglected here
as the variation in amplitude is 2 rad/s or 5% of the mean, and can
be approximated with small error as a line. The resultant
coefficients of dynamics, when measured from maximum p to maximum
q, varies between the models with A41 and B41 exhibiting similar
shifts of 33.0.degree. and 35.6.degree., respectively. Models C41
and D41 displayed substantially larger phase shifts of
131.7.degree. and 188.7.degree., respectively. An increase in phase
shift between p and q corresponds to an increase in the radius of
precession of the center of mass, r.sub.CG.
[0198] The results of the Fourier analysis are tabulated in Table
7.
TABLE-US-00007 TABLE 7 Coefficients of the Fourier series estimate
of TSA roll and pitch State w, rad/s a.sub.0 a.sub.1 b.sub.1
a.sub.2 b.sub.2 a.sub.3 b.sub.3 MSE {circumflex over (p)}.sub.A(t)
6.206 -0.323 +5.905 -7.398 +0.049 -0.229 -- -- 0.789 {circumflex
over (q)}.sub.A(t) 6.194 +0.257 +0.297 -6.677 -0.147 -0.240 -- --
0.024 {circumflex over (p)}.sub.B(t) 2.773 -1.872 +4.004 +3.147
+3.450 -12.10 -- -- 0.696 {circumflex over (q)}.sub.B(t) 6.185
+0.018 -0.224 -6.912 -0.128 -0.260 -- -- 0.225 {circumflex over
(p)}.sub.C(t) 4.059 -2.573 -4.163 +5.055 -1.399 -3.363 -0.331
+0.431 0.386 {circumflex over (q)}.sub.C(t) 6.334 +0.948 +1.273
+6.392 +0.179 +0.447 -- -- 0.091 {circumflex over (p)}.sub.D(t)
2.947 -2.462 +0.172 +4.173 +7.113 -4.525 +0.341 -1.170 0.118
{circumflex over (q)}.sub.D(t) 6.180 +1.349 -7.254 +1.627 -0.175
-0.460 -- -- 0.598
[0199] The subject study of the mechanical Samara A41, B41, C41,
D41 models elucidated the effects of the mechanical Samara wing
geometry on its autorotation and rotational dynamics, the
characterization of previously unobserved roll and pitch dynamics,
and a correlation between the different planform geometries of the
mechanical Samara models tested and a variation in the descent
velocity. Additionally, the radius of precession of the center of
mass r.sub.CG was found to vary by >500% between models, and a
linear relationship was established between the location of the
area centroid with respect to the center of mass, Y.sub.C, and the
vertical descent velocity, .sub.CG, which captures the trends
observed by varying planform geometry.
[0200] The study of the mechanical Samara models has been further
advanced into the design of the micro/nano scale mono-wing
rotorcraft presented in FIGS. 1A-1B, 15 and 16A-16D, that uses a
new airfoil derived from natural Samara chord geometry and mimics
the passive transit of the species of Samara (winged seed), Acer
diabolicum Blume. A simulated electronics payload of 4.5 grams was
molded into the model of the subject rotorcraft to demonstrate the
vehicles ability to safely deliver sensitive hardware by the
ballistic descent with consistent autorotation.
[0201] A substantial challenge in modeling the dynamics of
microscale flight is the general lack of knowledge of the complex
low-Reynolds-number flow regime inhabited. Additionally, the
vehicles are highly susceptible to wind gusts as a result of low
vehicle inertia. The complexity of the system can be reduced
substantially by identifying a linear model that describes its
reaction to forces imposed by a control input. A model description
of this nature lends itself well to modern control and state
estimation.
[0202] A large number of the models of the Samara mono-wing
miniature air vehicle have been built and studied with the purpose
of characterization of the flight dynamics and of control of the
subject robotic micro air vehicle 10 of the present invention.
Identification and error estimation of the vehicle dynamic model
has been accomplished with data collected by a visual tracking
system and usage of a frequency response-based system
identification package developed at the U.S. Army Aeroflight
dynamics Directorate located at Moffett Field, Calif.
Frequency-domain system identification is limited to linear models,
valid only for small perturbations about the trimmed condition.
[0203] The subject model structure is based on a reduction of the
longitudinal dynamics of a helicopter in hover to a linear system
described by stability derivatives. Cramer-Rao (CR) insensitivity
bounds describe the likely error resulting from multiple trials and
are used to validate the estimates of the identified parameters.
Additional validation of the linear model is done through
implantation of a linear controller, with feedback provided by the
visual tracking system.
[0204] Presented in FIGS. 1A, 15 and 16A-16D, Samaras I and II
models use unconventional and Samara-inspired planform geometry and
airfoil cross sections. These vehicles perform stable autorotation
and are capable of landing at terminal velocity without sustaining
any damage. In the event of motor failure, the vehicles gently
autorotate back to the ground.
[0205] The propeller(s) 28 of Samaras I and II are spinning in the
c.sub.y-c.sub.z plane and oppose applied torque about the c.sub.x
axis, providing additional roll stability. The configurations of
Samaras I and II models permit control of rotation rate, altitude,
and translation via the appropriate actuation of the wing servo
mechanism 18.
[0206] The design and construction of the Samara aircrafts used in
the subject experiments was made with the intent of providing a
stable vehicle that could be tested in a limited area. The
unconventional wing and body structure are the result of an
iterative design process that has produced on the order of 100
vehicles. The resultant vehicles are extremely damage tolerant as
they employ flexible structures that deflect upon impact,
effectively increasing the time over which the impact load is
applied to the vehicle. The configuration and dimensions of the
Samara I, II air vehicles are shown in FIGS. 16A-16D. Advantages
over traditional microscaled vertical takeoff and landing
configurations include passive stability, efficient autorotation,
low body drag, mechanical simplicity, low cost, high payload
capacity, and substantial damage tolerance.
Vehicle Design
[0207] All particular details of the subject design, including
specific dimensions, weight (mass), materials, and parts are given
herein as an example only of the models tested. However,
alternative design constituents are also contemplated in the scope
of the present invention.
[0208] The primary load-bearing structure of the vehicle was made,
for example, for 0/90-ply 0.025-thick carbon-fiber composite
laminate, with opposed parallel tension and compression members
mounted to the motor 26 and wing 12. In this configuration, the
structure provides a high degree of flexure in the c.sub.z
direction and a high degree of stiffness in the plane of rotation.
The angle at which the motor 26 is held provides protection from
ground impingement on takeoff and landing. The allowable range of
the angular deflection of the motor/propeller will be detailed
further herein.
[0209] The flight time of the Samara I in roughly 20 min, with a
0.025 kg, 480 mAh, 7.4 V, and two-cell lithium-polymer (LiPo)
battery, for a total vehicle mass (gross weight GW) of 0.075 kg.
The maximum gross takeoff weight (GTOW) of the vehicle was 0.125
kg, and the maximum dimension is 0.27 m.
[0210] The second, and smaller vehicle tested, called Samara II,
was designed and constructed in a similar fashion to Samara I.
However, the total mass was 0.038 kg, and the maximum dimension was
0.18 m. Table 8 details the mass breakdown of Samaras I and II, as
well as two hobby radio-controlled rotorcraft. The mass breakdown
is similar in the four vehicles. However, the Samara vehicle
benefits from a less complex, and therefore lighter,
propeller/rotor system and requires no transmission, as it directly
drives the propeller. This decrease in complexity creates a more
robust and reliable system.
[0211] D. Pines, et al., (Pines, D and Bohorquez, F., "Challenges
Facing Future Nitro-Air-Vehicle Development," Journal of Aircraft,
Vol. 43, No. 2, 206, pp. 290-305) found that the propulsion systems
of small-scale fliers typically exceed 60% of the total vehicle
mass, compared with a jetliner that has a propulsion system with a
40% mass fraction. The 20% savings at full scale is used entirely
for payload, as the Boeing 767 payload mass fraction is 29%,
compared with 9% for small-scale flight vehicles. The mass fraction
of the propulsion system of Samaras I and II without payload is 47
and 42%, respectively. Considering the vehicles` maximum GTOW for
this calculation, the mass fractions of the propulsion system drop
to 28 and 35%, respectively, with payload fractions of 40 and
21%.
TABLE-US-00008 TABLE 8 Weight data (in terms of percent of GW)
Hobby Hobby rotor- rotor- Parameter Samara II Samara I craft 1
craft 2 Mass 0.038 kg 0.075 kg 0.3 kg 1.8 kg Maximum GTOW 0.048 kg
0.125 kg -- -- Maximum dimension 0.18 m 0.27 m -- -- Percent GW
Propeller/rotor system 5.3 2.6 11.0 11.2 Tailboom assembly 2.6 3.3
8.0 9.1 Main motor (electric) 10.5 10.7 15.4 10.5
Fuselage/structure 26.3 27.6 7.0 15.1 Main transmission Direct
drive Direct drive 2.0 3.4 Landing gear 2.6 2.7 2.3 3.4 Control
system 18.4 16.0 5.7 18.3 Flight control avionics 7.9 4.0 29.4 2.4
Power source 26.3 33.3 19.2 26.6 Payload 0 0 0 0 Flight time 10 min
20 min -- --
[0212] Stability Properties
[0213] A substantial advantage of the Samaras I and II air vehicles
is that it they are passively stable systems. A simple qualitative
stability analysis of the Samara I in a steady hover or
autorotation illustrates this point. As shown in FIG. 15, in a
steady hover, the thrust from the propeller F.sub.p is balanced by
the drag from the body and wing F.sub.D.sub.W, resulting in a
near-constant rotational rate about its principal inertial axis
I.sub.z. The vertical force generated by the wing F.sub.L.sub.W
apposes the force of gravity F.sub.G, resulting in a net zero
vertical velocity. Alternatively, in autorotation, the resistive
torque of the wing drag is equal to the driving torque of the lift,
resulting in a net zero torque and vertical acceleration.
[0214] Consider the assumed motion yaw r=r.sub.0 and roll, pitch p,
qr.sub.0 in steady hover or autorotation. To investigate whether
the motion is stable or not (neglecting aerodynamic contributions),
a small moment is applied to the body, such that, after the moment
is applied, the resultant angular velocities are as follows:
p=.epsilon..sub.p (24)
q=.epsilon..sub.q (25)
r=r.sub.0+.epsilon..sub.r (26)
where .delta..sub.i (i=1, 2, 3) are infinitesimal quantities. To
determine the evolution of these perturbed angular velocities in
time, it is convenient to use the Euler equations as follows:
I.sub.z({dot over (r)}.sub.0+{dot over
(.epsilon.)}.sub.r)+(I.sub.x-I.sub.y).epsilon..sub.p.epsilon..sub.q=0
(27)
I.sub.x{dot over
(.epsilon.)}.sub.p-(I.sub.y-I.sub.z)(r.sub.0+.epsilon..sub.r).epsilon..su-
b.q=0 (28)
I.sub.y{dot over
(.epsilon.)}.sub.q-(I.sub.z-I.sub.x)(r.sub.0+.epsilon..sub.r).epsilon..su-
b.p=0 (29)
The change in angular velocities is small and, as such, allows
linearization of the preceding equations by eliminating quadratic
and higher-order terms in .epsilon..sub.i, yielding
I.sub.z{dot over (.epsilon.)}.sub.r=0 (30)
I.sub.x{dot over
(.epsilon.)}.sub.p-(I.sub.y-I.sub.z)r.sub.0.epsilon..sub.q=0
(31)
I.sub.y{dot over
(.epsilon.)}.sub.q-(I.sub.z-I.sub.x)r.sub.0.epsilon..sub.p=0
(32)
This implies .epsilon..sub.r is constant. The behavior of the
remaining angular velocities can be understood with eigenvalue
analysis. Assuming a solution of the form
.epsilon..sub.p(t)=E.sub.Pe.sup..lamda.t (33)
.epsilon..sub.q(t)=E.sub.qe.sup..lamda.t (34)
Next, the expansion can be introduced into the linearized
equations,
[ I x .lamda. ( I z - I y ) r 0 ( I x - I z ) r 0 I y .lamda. ] [ E
p E q ] .lamda. t = [ 0 0 ] ( 35 ) ##EQU00006##
The solution requires that the determinant of the coefficient
matrix be zero, yielding the characteristic equation
I.sub.xI.sub.y.lamda..sup.2-(I.sub.x-I.sub.z)(I.sub.z-I.sub.y)r.sub.0.su-
p.2=0 (36)
The solution is
.lamda. = .+-. i ( I x - I z ) ( I z - I y ) r 0 2 I x I y ( 37 )
##EQU00007##
, where .lamda. are the eigenvalues, or solutions which make the
determinant of the coefficient matrix zero. They are also called
the poles of the system. Two types of solutions are possible and
depend on the principal moments of inertia. If I.sub.x>I.sub.z
and I.sub.y>I.sub.z or if I.sub.x</I.sub.z and
I.sub.y<I.sub.z (characteristic of Samara I and Samara II), both
roots of the characteristic equation are imaginary. In the absence
of nonconservative forces, the system is marginally stable. The
inertial parameters of the Samara vehicles, as well as the
resultant eigenvalues, are listed in Table 9.
TABLE-US-00009 TABLE 9 Inertia properties, rotation rates, and
resultant eigenvalues for robotic Samaras I and II I.sub.x,
I.sub.y, I.sub.z, kg mm.sup.2 kg mm.sup.2 kg mm.sup.2 r.sub.o,
rad/s .lamda., rad/s Samara I 248 562 797 80.5 .+-.0 + 77i Samara
II 35 98 122 76 .+-.0 + 59i
Experimental Setup
A. Visual Tracking System
[0215] The position and orientation of each vehicle was collected
at a rate of 500 Hz, using a VICON visual tracking system. The
system collects two-dimensional (2-D) images of the subject, which
is fitted with retroreflective markers. The VICON system strobes
light at the frame rate of the camera. The light incident on the
surface of the marker returns to is source, reducing errors
commonly caused by interference. The light returned to the lens
allows for a quick computation of the centroid of the marker.
[0216] The three-dimensional (3-D) position was obtained from a
least-squares fit of the 2-D camera observations. The error in the
system was determined by the calibration of the system, which was
performed before any flight data were taken. The noise observed in
a given data set was consistent.
[0217] During a flight test, the tracking system used eight cameras
to track the 3-D position of three retroreflective markers placed
on the Samara wing. Each marker was spherical or shaped as a disk,
with a diameter of 5 mm. The 3-D shape of the marker allows for
better tracking by the VICON system.
B. Telemetry Synchronization
[0218] Pitch input was measured by two methods, both onboard and
offboard the vehicle. The state of the actuator 54 of the OFF-board
controller 22 (shown in FIGS. 1A and 1B) was measured offboard the
Samara on an identical system receiving commands from the same
transmitter. Two markers were placed on an arm attached to the
offboard actuator to track the input to the vehicle. During a
flight test, the Samara vehicle and the offboard actuator were
simultaneously tracked, allowing the angular displacement measured
on the ground to be correlated to the motion of the Samara vehicle,
both of which were synchronized in time.
C. Vehicle Inputs
[0219] It is advantageous to track the wing pitch angle via the
offboard system, as it provides the ability to track the collective
pitch inputs without the influence of the forces on the vehicle.
The onboard method includes measuring both pitch angle .theta. and
coning angle .beta. via the markers placed on the wing. It is
worthwhile to compare the onboard and offboard measurements, as the
onboard angles are influenced by the aerodynamic and centrifugal
forces acting on the vehicle.
[0220] The wing pitch angle .theta..sub.0 of the vehicle in flight
exhibits a once-per-revolution displacement. This variation results
in a cyclical change in the coning angle .beta. and heave velocity
not captured by the offboard measurements. In the absence of
aerodynamic forces, the wing would assume an arbitrary orientation.
However, in the presence of aerodynamic forces, a flapping moment
is applied to the Samara vehicle body, causing the wing to precess
to a new orientation, restoring equilibrium to the system. Nothing
was presumed to be known about what forces or deflection angles
were generated, given a change in the actuator; therefore, all
control inputs are normalized. The input command is given by
.theta..sub.0 for collective input and is normalized, such that
.theta..sub.0.epsilon.[-1, 1]. The forces acting on the wing in
flight, as well as the definition of the coning angle, are detailed
in FIG. 17.
D. Attitude Determination and Attitude Representation
[0221] Altitude determination and representation have been
described by Eqs. (2)-(17) detailed in previous paragraphs.
E. Kinematic Output
[0222] The VICON-obtained estimates are exceptionally low-noise as
compared with commercial-grade onboard attitude estimation sensors.
The position noise variance shown in Table 10 was estimated by
recording data while not moving the vehicle. The low-noise presence
in the position estimate allows the inertial position to be
numerically differentiated to yield inertial velocity
estimates.
{ x . , y . , z . } T = .differential. .differential. t { x , y , z
} T ( 38 ) ##EQU00008##
The body-fixed velocities may be directly computed using the
direction cosine matrix representation of the orientation estimate
R.sub.BF and the inertial velocities as
{u,v,w}.sup.T=R.sub.BF{{dot over (x)},{dot over (y)}, }.sup.T
(39)
TABLE-US-00010 TABLE 10 Measurement characteristics Measure- ment
Symbol(s) Source Resolution Variance Unit Time t VPS 1.000 .times.
10.sup.-3 -- s Control input .theta..sub.0 VPS -- 7.8000 .times.
10.sup.-3 norm Position x, y, z VPS -- 0.613 .times. 10.sup.-3 m
Orientation .phi., .theta., .psi. VPS -- 7.800 .times. 10.sup.-3
rad Translational u, v, w VPS -- 0.251 .times. 10.sup.-3 m/s
velocity Rotational p, q, r VPS -- 1.200 .times. 10.sup.-3 rad/s
velocity
F. Open-Loop Flight Test Data
[0223] The first step in system identification is to pilot the
vehicle in a flight envelope where the dynamics of interest are
thoroughly excited. The vehicle was piloted within the capture
volume of the vision system while simultaneously collecting the
inputs and vehicle kinematics. The pilot excited the vehicle over a
wide range of frequency content to best determine the relationship
between input and output. For proper system identification, it is
important to collect flight data as open loop, since a closed-loop
feedback system would alter the natural dynamics of the vehicle.
The open-loop setup is shown in FIG. 18.
[0224] Typical portions from recorded open-loop data sets are shown
in FIGS. 19A-19B. The heave velocity w was found by applying the
central difference approximation to the vehicle vertical position
data collected by the VICON system. FIGS. 19A-19B also present the
comparison between the inputs given to the vehicle during one
flight test, as calculated both onboard and offboard the robotic
Samara air vehicle. Both onboard and offboard methods demonstrate
similar pitch inputs, but the onboard measurements display more
oscillations.
G. Closed-Loop Flight Test Data
[0225] Implementation of closed-loop flight was enabled by an
offboard feedback system. The ground control station setup is shown
in FIG. 20. During closed-loop flight, the position and orientation
of the robotic Samara air vehicle were tracked by the VICON visual
system, which sends the formation to a LabVIEW controller program
[LabVIEW, Software Package, Ver., National Instruments, Austin,
Tex., 2008]. The LabVIEW program takes into account the vehicle's
vertical position and heave velocity to create wing collective
commands that are sent through PIC-18F8722 microcontroller. The
programmable interface microcontroller in turn sends the commands
to the vehicle through a Spektrum transmitter.
Experimental Results
A. System Identification Method
[0226] In the identification process, the coherence function was
computed. This step provides a measure of the extent to which an
output is linearly related to the input over some frequency range.
The magnitude squared coherence is given by
.gamma. xy 2 ( .omega. ) .ident. R xy ( .omega. ) 2 R xx ( .omega.
) R yy ( .omega. ) ( 40 ) ##EQU00009##
An input/output pair with low coherence implies that either the
input has no effect on the output or the effect is nonlinear.
However, an input/output pair with high coherence implies the
relationship can be modeled well by a linear model, such as a
transfer function or state-space model. A coherence of 0.6 or above
for some useful frequency range is believed to be necessary for
accurate transfer function identification.
[0227] The magnitude-squared coherence for the input/output
relationship of Samara I, using the onboard actuator system for
input measurement, is shown in FIG. 21A. It can be seen that the
useful frequency for this input/output pair lies in the range of
0.3 to 10 Hz. The coherence and useful frequency range predicted by
the onboard measured .theta..sub.0 is equivalent to that of the
offboard measurement (FIG. 21A). The similarity of the two
predictions validates the hypothesis that offboard measurements of
.theta..sub.0 are capable of capturing the physics relevant for
system identification. The onboard measurement of .theta..sub.0 for
Samara II (FIG. 21B) demonstrates some high frequency behavior
above 55 rad/s and may be a result of the aeroelasticity of the
wing in flight.
[0228] Samara-I does exhibit lower correlation than Samara II below
5 Hz, most likely due to less excitation of Samara I in that
frequency range as compared with Samara II. Despite the lower
frequency content observed in the flight test of Samara I, all
three coherence plots demonstrate similar ranges for strong
relationships between input and output.
B. Open-Loop Control
[0229] The transfer function of the pitch-input-to-heave dynamics
was modeled as a first-order continuous-time process model:
G p ( s ) = K s - T pl = W ( s ) .THETA. ( s ) ( 41 )
##EQU00010##
, where k, s, T.sub.pe, w(s), and .theta.(s) may be found with
Nomenclature list presented herein. Given a flight data set with
sufficient coherence, as seen in FIGS. 21A-21B, the MATLAB.RTM.
system identification toolbox may be used to compute frequency
response-based system identification. The input and output data are
imported to the system identification GUI (Graphical User
Interface) where it is filtered to 100 rad/s using a fifth-order
Butterworth filter.
[0230] Table 11 presents the values identified for the subject
robotic Samara air vehicle for the collective-to-heave velocity
transfer function using data from both methods of measuring pitch
input. In comparing the two methods of identification, it is
important to note that both methods identify K (static gain) and
T.sub.pl (time constraint) to be on the same order of magnitude,
proving both methods have similar capabilities in capturing the
input/output relationship. The transfer functions of the computed
models are plotted in FIGS. 21A-21B and 22A-22B for Samara I and
Samara II models.
TABLE-US-00011 TABLE 11 Identified robotic Samara parameters Samara
I Samara I Samara II .theta..sub.0 Offboard Onboard Onboard K
-13.643 -24.689 -21.44 T.sub.pl -4.864 -3.814 -1.690
C. Error Analysis
[0231] A state-space model was created allowing for error
estimation, using the CR (Cramer Rao) bounds, and is represented
as
{dot over (X)}=AX+BU (42)
Y=CX (43)
, where {dot over (X)} is the derivative of the state vector, A is
a dynamics matrix, B is a control matrix, C is an output matrix, X
is a state vector, Y is a control output, and
U.sub.T=[.theta..sub.0], where .theta..sub.0 is a control input
(pitch), as found in the Nomenclature list presented herein. The
state-space model for this identification reduces to
{dot over
(.omega.)}=Z.sub..omega..omega.-Z.sub..theta..sub.0.sub..theta..sub.0
(44)
The CR bounds are theoretical minimum limits for the expected
standard deviation I in the parameter estimates. The following
conditions are suggested to represent the most valid parameter
estimates: CR.ltoreq.20% and .ltoreq.10%. The CR and percentages
were found using the Comprehensive Identification of Frequency
Responses software (CIFER). The parameter estimates and associated
error bounds of the identified state-space model are presented in
Table 12. The validity of the identified parameter estimates is
hereby demonstrated, as all parameters meet the conditions
specified.
TABLE-US-00012 TABLE 12 Robotic Samara-identified parameter with CR
error estimates Term Value CR, % , % Onboard Samara I Z.sub..omega.
-6.382 10.04 4.231 Z.sub..theta..sub.0 -15.880 4.733 1.994 Offboard
Samara I Z.sub..omega. -4.303 9.413 3.808 Z.sub..theta..sub.0
-28.130 5.022 2.032 Onboard Samara II Z.sub..omega. -20.640 13.670
2.064 Z.sub..theta..sub.0 -1.501 12/840 1.939
[0232] The model computed from both on/offboard measurement of the
collective angle input is capable of capturing most of the
low-frequency inputs but may be seen to average higher-frequency
excitation. The model computed from the offboard measurement of the
collective angle input performs well at the lower frequencies but
tends to average the higher-frequency excitation The model exhibits
more overshoot than that of the model derived from onboard
measurements. The small differences in the performance of the two
methods of input measurement validate the ground-based input
observation method. A comparison of the poles identified by MATLAB
and CIFER is displayed in FIGS. 23A-23B for Samara I and Samara II,
respectively. The control derivative is a negative number as an
increase in collective pitch .theta..sub.0 results in an increase
in rotor thrust.
D. Heave Dynamics
[0233] The heave dynamics of the robotic Samara mono-wing aircraft
in hover are described by
{dot over (.omega.)}-Z.sub..omega..omega.=0 (45)
which has the analytical solution
.omega.(t)=.omega..sub.0e.sup.Z.sup..omega..sup.t (46)
Because the heave stability derivative Z.sub..omega. is negative,
the motion following a heave perturbation is a stable subsidence,
as shown in FIGS. 24A-24B for Samara I and Samara II, respectively.
For example, a positive heave perturbation will generate an upflow
through the robotic Samara rotor disk and increase thrust, which
acts in the negative direction of the c.sub.z-body axis. This also
implies that, in hover, the robotic Samara aircraft will have a
real negative pole, as shown in FIGS. 23A-23B. It is also possible
to obtain the expression for altitude loss due to a velocity
perturbation .omega..sub.0. For a robotic Samara in hover, .omega.=
and
z(t)=.intg..sub.0.sup.t.omega.dt+z.sub.0=.omega..sub.0.intg..sub.0.sup.t-
e.sup.Z.sup..omega..sup.tdt+z.sub.0 (47)
where z.sub.0 is the initial altitude. Integrating from {0, t}
yields
z ( t ) = z 0 - .omega. 0 Z .omega. [ 1 - Z .omega. t ] ( 48 )
##EQU00011##
from which the asymptotic value of altitude loss is
lim t .fwdarw. .infin. .DELTA. z = - .omega. 0 Z .omega. ( 49 )
##EQU00012##
The robotic Samara altitude change in response to a perturbation of
heave velocity is shown in FIGS. 24A-24B.
E. Heave Response to Pilot Input
[0234] Consider a step input of collective pitch .theta..sub.0=0.4.
After a change of variables, the heave dynamic equation can be
written as
{dot over (.omega.)}.sub.1-Z.sub..omega..omega..sub.1=0 (50)
where
.omega. 1 ( t ) = .omega. ( t ) + Z .theta. 0 Z .omega. .theta. 0 ;
.omega. . 1 = .omega. ( 51 ) ##EQU00013##
The analytic solution of the first-order differential equation
is
.omega..sub.1(t)=.omega..sub.1.sub.0e.sup.Z.sup..omega..sup.t
(52)
with
.omega. 1 0 = { .omega. + Z .theta. 0 Z .omega. .theta. 0 } t = 0 +
( 53 ) ##EQU00014##
For the robotic Samara in a steady hover, .omega.=0, which reduces
the solution of .omega..sub.1(t) to
.omega. 1 ( t ) = Z .theta. 0 Z .omega. .theta. 0 Z .omega. t ( 54
) ##EQU00015##
Thus, the heave velocity response to the input of collective pitch
reduces to
.omega. ( t ) = - Z .theta. 0 Z .omega. .theta. 0 ( 1 - Z .omega. t
) ( 55 ) ##EQU00016##
An example of the first-order character of the vertical speed
response to a step input of collective pitch is shown in FIG. 25.
This is a basic characteristic of the behavior of a robotic Samara
and is clearly identifiable in results obtained from mathematical
models and flight tests.
F. Closed-Loop Feedback Control
[0235] Feedback control is used to correct for perturbations in the
system in order to keep the vehicle at a reference condition. The
structure of the closed-loop system 60 is depicted in FIG. 26.
Precise attitude data are collected by the VICON motion capture
system. The commanded altitude of the Samara aircraft was
maintained by feeding back the error in position to a control loop
that contains the dynamics of the system and the actuator of the
off-board controller. The closed-loop system attempts to compensate
for errors between the actual and reference height of the Samara
aircraft by measuring the output response Y, feeding the
measurement back and comparing it to the reference value Y.sub.d at
the summing junction 62. If there is a difference between the
output and the reference, the system drives the aircraft's wing to
correct for the error.
[0236] A Proportional Plus Derivative Plus Integral (PID)
controller was chosen for feedback control of the robotic Samara. A
PID controller in FIG. 26 is described by the equation,
K ( s ) = K p + K d s + K i s ( 56 ) ##EQU00017##
, where s is the Laplace transform of the time domain equation into
frequency domain. The Gp(s) function of FIG. 26 is described by Eq.
41 presented supra herein. A PID controller feeds the error plus
the derivative of the error forward to the wing. The proportional
gain provides the necessary stiffness to allow the vehicle to
approach the reference height. The proportional gain improves the
steady-state error but may cause overshoot in the transient
response, whereas the derivative gain improves transient response.
The integral term is proportional to both the magnitude and
duration of the error in position, with the effect of eliminating
the steady-state error.
[0237] Using the ground control station setup shown in FIG. 20 for
closed-loop feedback control, several gain combinations were tested
in order to find the PID gains, providing the best transient
response to a change in reference height. FIGS. 27A-27B depict a
representative data set of a flight test with the implementation of
the PID controller for Samara I and Samara II, respectively, using
the gains presented in Table 13, demonstrating that the actual
height closely matches the reference height.
TABLE-US-00013 TABLE 13 PID gains for feedback control Gain Samara
I Samara II K.sub.p 0.211 0.344 K.sub.d 0.889 0.133 K.sub.i 0.028
0.020
[0238] The dashed line in FIGS. 27A-27B is the altitude specified
by the ground station. The solid line is the vertical flight path
of the aircraft in question. The change in altitude specified for
ascent and descent are the same and, for a linear controller, the
initial change in collective input is also the same. However, the
resulting heave velocity in ascent is half the value observed in
descent for both the Samara I and Samara II. The characteristic
overdamping in climb and underdamping in descent of Samaras I and
II are the result of the effect of gravity on the vehicle. In
climb, the input wing force is greater than and opposite the force
of gravity. In descent, the same input wing force is in the
direction of the force of gravity, resulting in a greater
acceleration.
[0239] The settling time T.sub.s of the Samara I for a climbing
maneuver is 1.03 s with no overshoot. A descending maneuver settles
to 90% of the final value within 1.45 s, with an overshoot of 22%.
The smaller Samara II reached 90% of its final value in 1.7 s, with
an overshoot of 60% for a descent maneuver. The settling time for a
climbing maneuver is 0.7 s, with a 4% overshoot. It can be seen
that the forces induced on the body from a change in collective
pitch are substantial when compared with the inertia of the
vehicle, as increases in heave velocity are quickly damped after
excitation.
[0240] Following the identification of the linear model, describing
the heave dynamics of two robotic Samara vehicles, e.g., Samara I
and Samara II, for use in future control and state estimation the
control model for the robotic Samara aircraft, including the
dynamics of a coordinating helical turn has been developed, as will
be presented infra herein.
[0241] The asymmetric and all-rotating platform requires the
development of a novel sensing and control framework. For this, the
general rigid body dynamics have been separated into rotor dynamics
and particle navigation, which were derived for a coordinated
helical turn flight path. The equations of motion have been used to
calculate the forces necessary for flight along a trajectory
recorded with a visual motion capture system. The result is a
framework for state estimation and control, applicable to scaled
versions of the robotic Samara.
[0242] As was presented supra herein, the layout of the subject
Samara aircraft includes two rigid bodies linked by a servo
mechanism allowing one rotational degree of freedom. The first
rigid body and main lifting surface resembles a scaled version of a
Samara both in planform geometry and airfoil cross section. The
second rigid body houses the electronics and motor/propeller unit
applying a torque to rotate the vehicle as required for flight. The
body fixed axis [c.sub.x, c.sub.y, c.sub.z] and Euler angles
[.phi..sub..omega., .theta..sub.0, .psi..sub..omega.] describe the
orientation of the vehicle, which is shown in FIG. 16D, along with
the dimensions of the vehicle.
[0243] Flight of a monocopter differs from full scale helicopters
as there exists no stationary frame of reference from which control
inputs can be applied, i.e. helicopter swashplate. Control of the
vehicle with once per revolution inputs requires knowledge of the
vehicle's orientation relative to the desired flight path, but
sensor packages capable of recording on-board flight data at the
rate necessary for this type of control are not commercially
available in the weight class required for use on micro/nano-class
vehicles. Instead, control algorithms development is based on state
information collected externally using a visual motion capture
system. This approach has been successful in identifying the pitch
and heave dynamics of similar vehicle which was described supra
herein. An approach to directional control which does not require
the once per revolution actuation or high frequency measurement of
vehicle orientation is discussed in detail further herein.
[0244] The obtained position estimates are exceptionally low noise.
The position noise variance was estimated by recording data while
not moving the vehicle. Inertial velocity estimates were calculated
by differentiation of the inertial position, using a central
difference scheme.
Flight Dynamics Model
A. Virtual Body Model
[0245] For the subject robotic Samara, in contrast to traditional
MAVs, the body orientation evolves over time, ranging from a steady
rotation rate about the .sub.z axis in hover to a more complex
pitching, rolling, flapping and rotating motion in other flight
conditions such as the translational flight condition addressed in
the current study.
[0246] To simplify the description, the "disk" described by the
motion of the wingtip over each revolution, or "tip path plane"
(TPP) is considered for further analysis of the motion of the
subject mono-wing aircraft. As defined in traditional rotor-craft
analysis, the TPP considered is one that discards the harmonic
motion higher than 1/rev, allowing a plane to be defined from the
surface. The aerodynamic lift force may be considered to act
perpendicular to the TPP.
[0247] To describe the dynamics of the Samara aircraft, a virtual
(rigid) body 66 connected to the disc center 68 of the disk 70 with
an ideal hinge 72 is considered, with its center of gravity (CG)
located directly below the disc center and with the mass of the
Samara aircraft, as shown in FIG. 28. No aerodynamic moments may be
transmitted across an ideal hinge, splitting the
position/orientation dynamics into rotor dynamics describing the
flapping motion of the wing, and positional dynamics of the Samara
aircraft to be described using the translational equations of
motion for the motion of a point mass acted upon by the rotor disc
forces.
[0248] The forward flight of the vehicle in question is most
conveniently formulated in a non-rotating frame of reference
attached to the virtual body. The orientation of the virtual body
forward velocity u is defined by the projection of the velocity
vector onto the [ .sub.x, .sub.y] plane so that translation may
only occur in the u direction and v=0. The heave velocity w is
parallel to the inertial .sub.z axis and is shown in FIG. 29. Also
shown is the equal and opposite definitions of aerodynamic
incidence .alpha., and the flight path angle .gamma. in relation to
the virtual body velocities, u, w. The cyclic blade flapping is
defined as the angle between the wing and the inertial plane [
.sub.x, .sub.y]:
[u,v,w].sup.T=[V cos .gamma.,0,-V sin .gamma.].sup.T (57)
B. Equations of Motion for a Flapping Blade
[0249] FIG. 30 defines azimuth angles of the wing
.psi..sub..omega., virtual body .psi..sub.cg, and virtual body with
respect to the wing .psi.. FIG. 31 defines coning angles
.beta..sub.1s, .beta..sub.1c with forces acting on an element of a
flapping robotic Samara aircraft's wing.
[0250] In steady hovering flight the coning angle
.beta.=.beta..sub.0=constant and is independent of .psi.. In
forward flight the cyclically varying airloads induce an additional
flapping response that varies about the azimuth .psi.. The
aerodynamic, centrifugal, and inertial forces acting on the robotic
Samara aircraft's wing determine the observed coning angle, FIG.
31. A positive movement is defined as one which acts to reduce the
.beta.. The centrifugal force M.sub.CF may then be written for an
element along the span as
d(M.sub.CF)=my.sup.2.OMEGA..sup.2.beta.dy (58)
, where y is the distance along the wing, and the inertial moment
about the flap hinge as
d(I)=my.sup.2{umlaut over (.beta.)}dy (59)
[0251] Additionally the aerodynamic moment is
d(M.sub..beta.)=-Lydy (60)
, where L is the lift of the wing.
[0252] The sum of the applied moments from the differential
equation describing the blade flapping motion. The flap equation
may be written as a function of azimuth angle instead of time,
where .psi.=.OMEGA.t results in the following transformation:
{dot over (.beta.)}=.OMEGA.{dot over (.beta.)} and {dot over
(.beta.)}=.OMEGA..sup.2{umlaut over (.beta.)} (61)
[0253] The equation of motion of the robotic Samara aircraft's
flapping wing reduces to
.beta. + .gamma. 8 .beta. . + .beta. = .gamma. 8 [ .theta. - 4 3
.lamda. I ] ( 62 ) ##EQU00018##
, where .gamma. is the lock number of the robotic Samara
aircraft,
.beta. . = .differential. .beta. .differential. t = .OMEGA.
.differential. .beta. .differential. .PSI. = .OMEGA. .beta. *
##EQU00019## .beta. = .differential. 2 .beta. .differential. t 2 =
.OMEGA. 2 .differential. 2 .beta. .differential. .PSI. 2 = .OMEGA.
2 .beta. ** ##EQU00019.2##
The lock number is a function of the aerodynamic and geometric
parameters listed in Table 14 and is computed as
.gamma. = pCl .alpha. cR 4 I b = 6.75 . ( 63 ) ##EQU00020##
[0254] Detailed numerical and steady state analytic solutions for
the flap equation in Eq. 62 have demonstrated good agreement with a
first order harmonic series. Harmonic analysis of the flap equation
allows a periodic solution of the form
.beta.=.beta..sub.0+.beta..sub.1s sin(.psi.)+.beta..sub.1c
cos(.psi.). (64)
The blade flapping throughout the u-turn is observed to be periodic
with respect to the azimuth angle .psi.. The periodic coefficients
describe the direction of force and may be seen to correlate with
both the velocity and acceleration of the virtual body in FIGS.
32A-32D. The .beta..sub.1c term influences the magnitude of u; and
the .beta..sub.1s term influences the magnitude
.parallel.V.sub.cg.parallel.. The coefficients .beta..sub.1s,
.beta..sub.1c are the out of plane flapping angles that describe
the orientation of the wing within the disk. The orientation of the
virtual body defines the roll and pitch angles to be
.phi.=.beta..sub.1s and .theta.=.beta..sub.1c, respectively. Thus
the flapping of the wing in forward flight describes the
instantaneous orientation of the virtual body which includes the
coning angle .beta..sub.0.
TABLE-US-00014 TABLE 14 ROBOTIC SAMARA WING PROPERTIES Measurement
Symbol Value Unit Air density p 1.225 Kg/m.sup.3 Mean chord c 3.5
cm Wing length R 18 cm Lift curve slope C.sub.l.alpha. 3.5 -- Wing
inertia I.sub.b 23.3.mu. Kgm.sup.3
C. Rigid Body Equations of Motion
[0255] The rigid body equations of motion are differential
equations that describe the evolution of the state variables
subject to applied forces. In body-fixed axes the sum of all
external forces applied to the center of gravity is
m{dot over (V)}.sub.cg+mS(.omega.)V.sub.cg=f (65)
, where m is the vehicle mass, V.sub.cg=us.sub.x+vs.sub.y+ws.sub.z
is the translational velocity of the center of gravity,
.omega.=ps.sub.x+qs.sub.y+rs.sub.x are the body-fixed roll, pitch
and yaw rates, f=f.sub.xs.sub.x+f.sub.ys.sub.y+f.sub.zs.sub.z are
externally applied forces, and S(.cndot.) is a skew operator.
[0256] The rotational dynamics are governed by the differential
equation
I{dot over (.omega.)}+S(w)I.omega.=.tau. (66)
, where .tau. is a vector of externally applied torques and I is a
diagonal inertia matrix arising from symmetries in the virtual
aircraft.
D. Coordinated Helical Turn
[0257] The flight path of the vehicle resembles a steady banked
turn such that {dot over (.phi.)}.sub.0 and {dot over
(.theta.)}.sub.0 are equal to zero. Additionally .gamma. the flight
path angle (>0 for climbing flight) is small so that sin
.gamma.=.gamma. and cos .gamma.=1. The kinematic equations are
then
p=-{dot over (.psi.)}.sub.cg sin .beta..sub.1c (67)
q={dot over (.psi.)}.sub.cg cos .beta..sub.1c sin .beta..sub.1s
(68)
r={dot over (.psi.)}.sub.cg cos .beta..sub.1c cos .beta..sub.1s.
(69)
, where {dot over (.psi.)}.sub.CG is a turn rate. Substituting the
derived velocities and modified kinematics into the force
equilibrium equations results in the following equations of
motion:
X=mg sin .beta..sub.1c+m({dot over (u)}+.omega.q-vr) (70)
Y=mg cos .beta..sub.1c sin .beta..sub.1s+m({dot over
(v)}+ur-.omega.p) (71)
Z=mg cos .beta..sub.1c cos .beta..sub.1s+m({dot over
(.omega.)}+vp-uq) (72)
, where [X, Y, Z].sup.T represent force equilibrium in the body
fixed coordinate frame, and g is acceleration due to gravity.
[0258] Flight tests conducted with the robotic Samara aircraft
provide a means of verifying the equations of motion. A portion of
a flight which fits within the constraints of the proposed
analytical model is shown in FIGS. 32A-32D.
[0259] The variation of {dot over (.psi.)}.sub.cg with the turn
radius r.sub.turn is observed to be linear for most of the trial,
where a small but linear change in r.sub.turn corresponds to a
large change in turn rate {dot over (.psi.)}.sub.cg. The final
portion of the figure shows the linear change in forward speed
derivative {dot over (u)} with respect to r.sub.turn.
E. Extension to Forward Flight
[0260] 1) Pure longitudinal motion: Consider now straight flight as
a special case of a coordinated turn, where {dot over
(.psi.)}=p.sub.0=q.sub.0=r.sub.o=.beta..sub.1s=0. The equation of
motion along the s.sub.x-axis for forward flight may be written as
the combination of a nominal condition (represented by [ ].sub.0)
and a small perturbation .DELTA.[ ] as:
X.sub.0+.DELTA.X-mg.left brkt-bot.
sin(.beta..sub.1c.sub.0)+.DELTA..beta..sub.1c
cos(.beta..sub.1c.sub.0).right brkt-bot.=.DELTA.{dot over (u)}
(73)
Setting all perturbation quantities to zero .DELTA.[ ]=0 yields the
force equilibrium along trimmed forward flight:
X 0 m = g sin .beta. 1 c 0 ( 74 ) ##EQU00021##
[0261] 2) Perturbation equations: The trimmed forward flight Eq.
(74) can be subtracted from the linearized force equilibrium Eq.
(73) leading to a description of small perturbation motion about
the equilibrium condition as:
.DELTA. u . = .DELTA. X m - g .DELTA. .beta. 1 c cos .beta. 1 c (
75 ) ##EQU00022##
Separating out the linear effects of the longitudinal variables [u,
.omega., .beta..sub.0, .beta..sub.1c] facilitates development of a
canonical linear control model, and may be written as:
.DELTA. X m = X u .DELTA. u + X .omega. .DELTA. .omega. + X .beta.
0 .DELTA. .beta. 0 + X .beta. 1 c .DELTA. .beta. 1 c + X .theta.
.DELTA. .theta. , ( 76 ) ##EQU00023##
where
X.sub.[.cndot.]=(1/m).differential.X/.differential.[.cndot.].
[0262] The time-invariant linear system may now be expressed in
state space form:
{dot over (x)}=Ax+Bu (77)
where
x=[.DELTA.u,.DELTA..omega.,.DELTA..beta..sub.0,.DELTA..beta..sub.1c].sup-
.T (78)
and
u=[.DELTA..theta..sub.0] (79)
written in matrix form as:
[ A ] = [ X u X .omega. 0 0 0 0 0 Z .OMEGA. Z .beta. 0 0 0 0
.OMEGA. .OMEGA. .OMEGA. .beta. 0 0 0 0 .beta. 0 .OMEGA. 0 0 .beta.
1 c u 0 .beta. 1 c .OMEGA. .beta. 1 c .beta. 0 0 ] ( 80 )
##EQU00024##
and
[B]=[X.sub..theta..sub.0,Z.sub..theta..sub.0,0,.beta..sub.0.theta..sub.0-
,0].sup.T. (81)
Experimental Results
[0263] Lateral directional flight was recorded in the laboratory
for a flight path consisting of an initial trim state and a
perturbation about the trim, shown in FIG. 33. In general, the turn
radius is inversely proportional to the collective pitch of the
wing. The Samara aircraft travels in the opposite direction of the
motion that would be induced by an impulsive collective input
applied at that instant. A non-impulsive, sustained input changes
the turn radius of the flight path such that an alternating series
of large and small turn radii can steer the vehicle in a specific
direction.
[0264] The resulting velocity components (forward u, vertical w),
rotation rate .OMEGA. (shown in FIG. 33B), and resultant flight
path and the radius of curvature (shown in FIG. 33C) are presented
as they vary with the input pitch .theta..sub.0 and throttle
F.sub.p (shown in FIG. 33A). The first 0.5 s of flight correspond
to a near constant u and near zero w. At the time of the u-turn,
1-1.5 s, there is an increase in the vertical velocity. The
increase is correlated because a collective pitch increase used to
change the heave velocity, is also used to change the flight path
direction. The flight data shown in FIGS. 33A-33C were used to
perform system identification using algorithms implemented in a
MATLAB toolbox called System Identification Programs for AirCraft
(SIDPAC), detailed in [V. Klein and E. Morelli, Aircraft System
Identification, AIAA, 2006].
[0265] Guided by analytical modeling, modified step-wise regression
was used to determine the model structure using the statistical
significance of measured states. This model structure was chosen to
maximize the model fit using regressors with a significant
partial-f ratio, while minimizing the parameter estimate error
bounds. A two-step procedure using the equation-error method,
followed by the output-error method, was used to estimate the
stability derivatives in the model.
[0266] The equation-error method performs a linear estimation at
the acceleration level, which has a deterministic solution that is
cheap to compute. The output-error method, widely regarded as more
accurate, performs a nonlinear estimation at the level at which
measurements were taken. This method requires an iterative
numerical solver, but initial guesses using an equation-error
estimate typically converge quickly.
[0267] Parameter estimates and standard errors, corrected for
non-white colored residuals, are given in Table 15.
TABLE-US-00015 TABLE 15 Parameter Estimates and Standard Errors
Parameter Equation-Error Output-Error .theta. {circumflex over
(.theta.)} .+-. s({circumflex over (.theta.)}) {circumflex over
(.theta.)} .+-. s({circumflex over (.theta.)}) x.sub.u +0.4165 .+-.
0.1753 +0.8978 .+-. 0.3416 x.sub..omega. +3.7378 .+-. 0.5694
+1.6108 .+-. 0.4431 X.sub..theta..sub.0 -7.9875 .+-. 21.055 +114.35
.+-. 25.827 Z.sub..OMEGA. +0.2538 .+-. 0.0164 +0.2237 .+-. 0.0318
Z.sub..beta..sub.0 -57.323 .+-. 12.301 -17.695 .+-. 23.318
Z.sub..theta..sub.0 -14.718 .+-. 13.399 +38.942 .+-. 19.449
.OMEGA..sub..OMEGA. -1.3358 .+-. 0.2084 -2.2649 .+-. 0.5876
.OMEGA..sub..beta..sub.0 +1001.5 .+-. 63.791 +1231.1 .+-. 219.49
.beta..sub.0.OMEGA. -0.0081 .+-. 0.0008 -0.0063 .+-. 0.0008
.beta..sub.0.theta..sub.0 -1.8922 .+-. 0.3399 -2.3455 .+-. 0.3158
.beta..sub.1c.sub.u -0.5881 .+-. 0.2122 +0.5628 .+-. 0.2050
.beta..sub.1c.OMEGA. -0.1145 .+-. 0.0325 +0.0872 .+-. 0.0540
.beta..sub.1c.beta..sub.0 +24.323 .+-. 7.7405 -22.543 .+-.
15.787
[0268] Model fits to the perturbation data sets are shown in FIGS.
34-35 for the equation-error and output-error methods, where the
measurements are plotted with a solid line and the model outputs
are plotted with a dashed line.
[0269] An acceptable fit was achieved between the analyzed data and
many of the model structure and parameter estimates. The
equation-error results had model fits of 0.94, 0.94, 0.91, 0.56,
and 0.59 for matching {dot over (x)} measurements and fits of 0.85,
0.97, 0.92, 0.83 and 0.12 for matching x measurements. The
equations describing the flap dynamics had low model fits for both
methods. Several parameters were estimated consistently by the two
methods, lying within two standard deviations of each other.
[0270] Significant insight may be obtained from the estimates that
did not match well or had large error bounds. Much of the
inconsistency may be attributed to limited excitation present in
the flight data, but it also reveals an important characteristic
about the analyzed flight data. For example, the finding that the
stability derivative X.sub.u is positive indicates a rare case of
forward speed instability. Forward speed instability is normally a
localized result and not found in a general flight dynamics result.
In this case, the portion of linearized flight used in the
equation-error output method fit included a portion of descending
flight, during which the robotic Samara (a global stability)
exhibits a local instability with respect to forward speed, because
arresting a forward velocity requires the rotor disc first be
inclined to oppose the motion. The estimated error in this term
reflects the fact that the stability derivative was required to fit
a more general flight case, but is biased toward descending
flight.
[0271] In general, the results of the experimentations helped in
identifying the Samara aircraft flight dynamics. The nonlinear
Euler equations were used to describe the rigid body dynamics of
the vehicle in a steady turn. The rotating wing motion was treated
as simple harmonic oscillator and coupled to the virtual body
equations of motion which combine to form a set of seven nonlinear
differential equations.
[0272] The equations of motion for the steady turn were extended to
forward flight and linearized about a trim state, resulting in five
linearized small perturbation equations in state space form. Flight
tests provided high accuracy position information that was reduced
to wing flap angles and virtual body velocities.
[0273] This information was used to specify a flight condition that
fits within the limits of the derived model allowing for estimation
of the parameters defined in the A and B matrices (Eqs. 80-81).
Several linear relationships were shown to exist including
[r.sub.turn,{dot over (u)}], and [r.sub.turn{dot over (.psi.)}],
The steady turn discussed here has been observed in scaled versions
of the robotic Samara aircraft. Therefore the open-loop control
demonstrated and analyzed is considered to be appropriate for
similar vehicles of reduced size with limited sensing and actuation
capabilities.
[0274] Based on the experimental results and analytical study,
including study of effects of the planform geometry on mechanical
Samara wing's autorotation efficiency and rotational dynamics,
study of pitch and heave control of the subject miniature
(micro/nano) mono-wing Samara aircraft, as well as its dynamics
about a coordinated helical turn, the novel mono-wing miniature
rotorcraft that mimics passive transit of the species of Samara
(winged seed) has been built and its flight has been controlled in
accordance with the model and equations described supra.
[0275] Summarizing the data presented supra herein, the direction
of travel of the Samara aircraft corresponds to the tilt of the
tip-plane. A tilt that results in the forward motion of the Samara
aircraft, .beta..sub.1c, the tilt resulting in the left/right
motion, .beta..sub.1s, and the mean flap angle, .beta..sub.0, are
defined in FIG. 31. The relationships between .beta., .beta..sub.0,
.beta..sub.1s and .beta..sub.1c are described by Eq. 64 presented
supra herein. Values recorded in flight for flap response with
.beta..sub.0, flap response with .beta..sub.1c and .beta..sub.1s,
and flap response with mean coning angle .beta..sub.0 are presented
in FIG. 36A, and the values of .beta..sub.0 and .beta..sub.1c for
forward flight are presented in FIG. 36B.
[0276] The primary challenge in controlling lateral directional
flight is to know where the vehicle is pointed and when to actuate
the control to steer it. To implement precise directional control a
1-per revolution actuation of the collective pitch angle
.theta..sub.0 is required to tilt the tip path plane in the
direction of a desired motion (in helicopter terminology this
control is called cyclic pitch control). The orientation of the
vehicle must be estimated on-board or off-board and send the
vehicle to implement this type of control. The estimation may be
done with an analog MEMS tri-axis magnetometer and two tri-axis
MEMS (microelectromechanical systems) accelerometers.
[0277] To implement the directional control with cyclic pitch the
system is treated as a simple harmonic oscillator. Ignoring damping
effects which influence the phase lead/lag from input to output,
the system responds to an input with a response 90.degree. out of
the phase.
[0278] For a cyclic input consisting of the following signal:
.theta.=.theta..sub.0+.theta..sub.1c cos .psi.+.theta..sub.1s sin
.psi. (82)
[0279] The wing flap response will be:
.beta.=.beta..sub.0+.theta..sub.1c cos(.psi.-.pi./2)+.theta..sub.1s
sin(.psi.-.pi./2) (83)
[0280] Referring to FIGS. 37A and 37B, within the allowable range
of the pitch (-30.degree./+30.degree.) shown in FIG. 37B, and
depending on the throttle, F.sub.p, and the input control
collective pitch .theta..sub.0, the miniature mono-wing air vehicle
operates either in the autorotation mode, performs climb or descent
motion, or operates in steady hover mode (as shown in FIG.
37A).
[0281] By providing input control signals presented in Table 16,
the aircraft in question is maneuvered and controlled.
TABLE-US-00016 TABLE 16 Commands for desired flight mode Command
.theta..sub.0 .theta..sub.1c .theta..sub.1s F.sub.p Hover 0 0 0 +
Autorotate - 0 0 0 climb + 0 0 + descend - 0 0 +, 0 forward 0 + 0 +
backward 0 - 0 + Left + S.sub.y direction 0 0 + + Right 0 0 - +
[0282] There are allowable ranges of angular deflections defined
for various components of the miniature mono-wing air vehicle
described herein. Referring to FIG. 38, representing the fuselage
range of acceptable angles, the arm 74 of the fuselage connecting
the motor 26 to the wing 12, and the arm 76 extending between the
battery and the wing, have the same range of acceptable angles
(.+-.60.degree.) for stable controlled flight.
[0283] Referring to FIGS. 39A and 39B demonstrating the
motor/propeller range of acceptable angles, the motor/propeller
acceptable angles for stable controlled flight range from a
positive 45.degree. (angling the thrust vertically) to -90.degree.
(angling the thrust to oppose gravity), as shown in FIG. 39A. The
motor/propeller acceptable angles for stable controlled flight
range from -45.degree. (angling the thrust towards the center of
gravity) which tends to increase hover stability to +45.degree.
(angling the thrust to away from the center of gravity) which tends
to decrease hover stability by causing an increasingly larger
circular path to ensure, as shown in FIG. 39B.
[0284] Referring to FIG. 40, an alternative embodiment is presented
with the propeller 28 positioned in opposite direction than the one
shown in FIG. 1A. Also, as shown in FIG. 41, a double
motor/propeller setup is envisioned which has more power and can
lift more pay load. The double motor/propeller setup has the added
bonus of single motor out operability. The propellers in this
embodiment may spin in either the same or opposite directions as
there is no need to cancel the torque from the propellers.
[0285] FIG. 42 is a schematic representation of the acceptable
range of ranging from -60.degree. to +60.degree.. This range may be
added before flight and is called a pre-cone angle.
[0286] Referring to FIG. 43, showing the allowable range of angular
deflection for the servo mechanism in flight, the damped motion of
the wing prevents damage upon impact and functions as a lead-lag
hinge. For this, the rubberized superglue and shock absorbing foam
are used in coupling of the servo mechanism to the fuselage and to
the wing.
[0287] The vertical thickness of the fuselage member 14 is the
parameter influencing the aerodynamical property of the vehicle
during its rotation around the end of the wing. If by adding
components an increase in thickness is required, it is preferably
added closest to the center of rotation, or the center of mass, as
this will minimize the drag apparent on the vehicle. Any thickness
of the fuselage and the drag it produces must be overcome by the
propulsive device to keep the vehicle spinning and flying. The
limit on thickness is one for which the propulsive force is too
small compared to the drag to keep the vehicle aloft.
[0288] The subject nano-micro mono-wing Samara aircraft
demonstrated excellent maneuverability, controllability, efficacy
in power consumption, steady hovering, and low descent velocity in
autorotation regime, thus making this vehicle ideally suitable for
operation in confined environments with high level of autonomy,
which is required in the reconnaissance mission. Being equipped
with a camera, the subject miniature rotorcraft can collect
detailed panoramic images due to the unique manner of motion in
flight, e.g., fast spinning about the end of the wing with a
frequency of about several rotations per second. The micro/nano
scale dimensions and light weight in the range of several tens of
grams, makes the vehicles in question easily deployed in an area of
interest at low cost.
[0289] Although this invention has been described in connection
with specific forms and embodiments thereof, it will be appreciated
that various modifications other than those discussed above may be
resorted to without departing from the spirit or scope of the
invention as defined in the appended claims. For example,
equivalent elements may be substituted for those specifically shown
and described, certain features may be used independently of other
features, and in certain cases, particular applications of elements
may be reversed or interposed, all without departing from the
spirit or scope of the invention as defined in the appended
claims.
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